Primality and Almost-Primality of Certain Polynomial Values
Created | Updated Oct 17, 2009
This Entry lists polynomials related to each other through their having a common evaluation for differing values of the variable, each polynomial
essentially being that which arise from considering this common evaluation--8413346833 in base ten--in different bases. After each polynomial
is the beginning of the list of values of the variable which generate a prime or an almost-prime (product of two primes), with those generating
primes marked by an asterisk. The bases that generate each polynomial are shown for convenience.
Base 2: x^32+x^31+x^30+x^29+x^28+x^26+x^24+x^22+x^21+x^20+x^19+x^16+x^14+x^13+x^12+x^11+x^10+x^4+1
1*,2*,7*,16,29,30,31,32,33,36,37*,43,51,64,70,72,80,100,103*,120,148*,150,170,192*,201,204*,224,241*,
246,247,254*,260,261,269,271,275,292*,311,315,318,344,346,352,359,365*,379,380,386,432,436,444,
448,453,456,464,470,483,486,505,518,530,531,534,546,551,556,563,565,574,591*,593*,622,651,658,
681*,721*,729,733,752*,768,771,781,787,793,794,796,805,828,843,866,868*,870,871,878,885,887*,897*,
904*,906,908,923,924*,927,939,950,973,982,983*,988,990,1003,1071,1094,1101,1115,1163,1164,1167*,
1181,1197,1216,1221,1232,1242,1270*,1279,1281*,1317,1318,1323,1326,1331,1335*,1360*,1372,1417,
1421,1456,1505,1506,1517,1519,1531*,1538,1558,1563,1571,1584*,1589*,1594,1611,1626,1641,1652,1699,
1703,1767,1772,1812,1813,1843,1848,1858,1863,1875,1885,1912,1927,1943,1951,1970,1972,1988,2009,
2018,2030*,2051,2060*,2067,2086,2087*,2089,2105,2115,2121,2122,2136*,2140,2143,2150,2165,2177,2182,
2191,2193,2250,2254,2255,2261,2268,2316,2317,2325*,2332,2362,
Comment: Only four primes or almost-primes through 28, then five consecutive; -1 only prime generator among negative x until -84.
Base 3: 2x^20+x^19+2x^17+x^15+x^14+x^10+2x^9+2x^7+2x^6+x^4+x^3+2x+1
1*,3*,4,7*,10*,12*,13*,15*,22,31,36*,37,41,42,48,52,54*,60,61,64,69,73,75,76,79,88*,89,
90*,96,99,111*,113,121*,129,132,135,139*,147,148,151,159,163,169*,178,184*,205,207,225*,
230,235,257,258*,262*,265,268,274,285*,289,291,297,304,334,340,359,363,372*,375,381,382*
Comment: As many primes as possible from 10 through 15, and primes at 88 and 90 with 3 times a prime at 89.
Base 4: x^16+3x^15+3x^14+x^13+x^12+x^11+3x^10+2x^9+x^8+x^7+3x^6+3x^5+x^2+1
1,3,4*,7,10*,15*,25*,28*,33,40,42,45,50,58,67,69,72,75*,78*,79,82*,85,90,93,102*,103,106,
108,109,114,117,118,120,127,132,142,145,148,163,168,169,177,208,219*,220*,222*,234*,242,
246,247,262,278,283*,285,288,295,309,325,330,332,336,345,349,358,369,372,375,384*
Comment: x=51 yields 5^9 times a prime, and, after a gap beginning with a prime at x=102, there are primes at x=219, 220 and 222.
Base 5: x^14+x^13+4x^12+2x^11+x^10+2x^9+3x^8+4x^6+4x^4+4x^3+3x^2+x+3
1,2,5*,8*,16,17,27,29*,35*,41,44,50,63,66,70,80,83,98,125*,128,131*,134*,158,170,176,
178,187,190,200,203,205,206,216,223,248,260*,275,290*,299,302*,308,314*,329,341*,344,
347,353,360,380,383,387,395,398,406,431,440,444,454,461,464,486,491,500,512,531,533*
Comment: Primes at 125, 131 and 134, with prior back at 35 and next not until 260; big gap 341-533.
Base 6: 3x^12+5x^11+x^10+5x^8+3x^6+x^5+x^4+5x^3+4x^2+4x+1
1,4,6*,15,18*,27,39,42,48*,60*,62,63,68,75,84,96,99,104,106,109,126,130,134,151,152,162,174,180,193,
201,208,231,238,246*,255,274,285,286,291*,294,305,309,330,333,334,348*,351,357,378,393,406,411,412,
427,442,456,465,468,473,483,504,516,531,550,554,561,564,567,568,582,627,636,642,643,663,678,690*
Comment: Primes rare, even considering the necessary condition that 3 divides x.
Base 7: 4x^11+x^10+5x^9+3x^8+3x^7+x^5+6x^4+6x^2+5x+3
1*,2*,5,7*,8,10,11*,16*,17,19,25,28,34,36,40*,44*,47*,50,55,56*,59,62,67,71,73*,80,81,82*,86*,90,95,
99,101,105,106,113,122,125*,127,154,157*,158*,170,176,185,187,189,191,193*,197,200,205,207,208,209,
223,224,227,233,235,241,242*,245*,248,249,254,260,265,268,269,272,275,276,278,288,295,296,311*,319,
320,325,334,335*,338,340,341,343,347,359,362*,364*,365,368,374,387,391*,394,396,397*
Comment: Some clustering of primes, with almost-primes also seeming to come in swarms.
Base 8: 7x^10+6x^9+5x^8+3x^7+6x^6+2x^5+7x^4+6x^3+2x+1
2*,8*,12,14,15*,18,23*,24*,29,33,42*,45,47*,53*,54*,57*,65*,68,74,77,78,80*,81,87,92,98,
107*,108*,113,117,119,120,128*,143,144*,147*,152,158*,161,165*,168,183*,185,203,206,207,
209,219*,227*,239,240*,242*,243,252,257*,260,264,284,287,308*,312,315,318,330,339,341,345,
362,378,383,392*,395*,399,407,410,414*,419,420*,425,438*,443,449,450,452,453*,455,465,470*
Comment: Early large cluster of primes, with clustering of both primes and almost primes continuing.
Base 9: 2x^10+3x^9+6x^8+4x^7+x^5+6x^4+8x^3+x^2+3x+7
1*,3*,4,5,6,9*,10,12*,15,16,19*,22*,25,26,27,30,31,33,34*,36,37,39*,40,45,46,58*,66,75,76,78,
81,82,85,94,99,103,109,120,129,139,142*,144,148,152,160,166,180,186,190,193*,195,214,216,226,
237*,250*,256*,267,274,276*,278,282,283,292,295,304,306,319,323,324,325*,337,339*,346,349,358,
367*,373,408,414,423,427,430*,433,438,439,447,448,457*,459,460,462,463,466,471,480,484,492,505*
Comment: String of almost-primes with just one prime, slight evidence of clustering.
Base 10: 8x^9+4x^8+x^7+3x^6+3x^5+4x^4+6x^3+8x^2+3x+3
1*,2*,3,4*,8*,9,10*,11,14,16*,22*,23,28,31,35*,41,46,47,49,59,61,62*,64*,68,70,73,76*,78,79,83,
85*,92*,94*,99,104*,106,107,108,118*,126,137,139,140,148,151,155,157,160,164,170,171,176*,177,
179,184,190,195,196*,197,199,200,205,206,208,218*,221*,223*,224,225,238*,241,244,247*,248,257,
265,268*,275,277,284,295,299,307,314,316,319,332,338*,344,349,350*,354,364*,368,374,377,380,382*
Comment: Nothing remarkable aside from a little clustering of primes.
Base 11: 3x^9+6x^8+2x^7+8x^6+x^5+2x^4+10x^3+1
1,2,3*,4,9,11*,15,17,20,21,28,29,30*,33,36,39,43,44,48,50,53,59,63*,66*,68*,69,80,90,99,101*,103,
104,105,110,114,119,124,129,134,140,146*,149,153,156,158*,159,164,168,169,174,184,185*,198*,204,
206*,207,213,220,224,228,230,231,233,236,251*,263,264*,267,269*,270,273,281,288,290,294*,299,
305*,306,308,318*,321,323,324*,325,326,339,345,357,358,360,362,365,374,378,379,381,383,384*
Comment: Some clustering, but not very notable.
Base 12: x^9+7x^8+6x^7+9x^6+7x^5+4x^4+2x^3+11x^2+8x+1
2*,4,6,10,12*,14*,16,20*,24*,26,28,32*,38*,42,44,46*,58,60*,66*,68*,70*,72*,86,94*,96,98*,102,104,
110,116,118*,122,138,160,164,168*,174,182,190,196,208,210,214,222,228*,230,240,242,248,252*,256*,
262,264*,266*,268,270,272,274,276,278*,284,290,296,298*,306,308,312,322,336,346,350,356*,366,384*
Comment: Four consecutive primes (66-72) and nine consecutive primes or almost-primes (262-278), odd numbers not considered.
Base 13: 10x^8+4x^7+x^6+7x^4+8x^3+7x^2+4x+8
1,3,9,11,13*,15,17,21,49*,51*,53*,55,57,65,69,75*,81,91*,93,95*,97,103,115*,117,119,129,133*,
137*,139*,143*,145,157,167,171,173*,181,187,189,191,213*,215*,217,219,229,231*,235,237,249,
255,259,273,275,283*,285,289,297,299,307,317*,321*,325,329,335*,341*,343*,347,353,363,371,375*
Comment: Substantial gaps and strings, and some clustering of primes.
Base 14: 5x^8+9x^7+11x^6+5x^5+4x^4+4x^3+6x^2+13x+3
10,14*,26,38,44*,64,74*,80,104*,164*,194,200,212,254,264,304,356,360,470,494,504,506,520,524*,614,660,694,710,
818,840,884*,930,944*,980,1010,1034,1064,1184,1240,1304,1352,1448,1454,1520,1538,1550,1560,1574,1594,1630,1634,
1700,1724,1790,1814,1820,1844,1890,1934,1954,1970,1974,1988,1994,2000,2012,2024,2120*,2135,2150*,...,2330*,...,2900*
Comment: Necessarily very sparse, and, after an enormous gap in primes, there are two primes around the first odd almost-prime generator.
Base 15: 3x^8+4x^7+3x^6+9x^5+4x^4+8x^3+7x^2+8x+13
1*,2,3,4,5,6*,7,8,9*,10*,11,14,15*,16*,19,21*,23,30,31,34*,40*,42,49,50*,51,54,55,56,61,69,71*,76*,
79,85,86*,89,96*,100*,101,103,111,114*,115*,121,124*,126,127,129,132,134*,135,136,139,140,142,
154,157,164*,167,170*,177,180*,181,185*,191,195,196,199,201,209,215*,222,224,226,229,236,241*
Comment: High density of primes and almost-primes through x=23, consecutive primes at 114-5.
Base 16: x^8+15x^7+5x^6+7x^5+9x^4+7x^3+12x^2+x+1
1,2*,3,4,5,6,9,10,13,14*,16*,20,21,23,25,28,29,30,31,32*,38*,40*,42,44,46*,48,49,50,52*,54*,56*,57,58*,60,
62,64*,66,67,72*,73,76,80*,86*,92*,94*,96*,98,100,103,104*,106,113,114*,118,120,122,123,124,126,128,132*,
136,143,144,148,150,152,162,164*,166*,168,171,173,174*,178,179,180,184,188,192,200*,201,202,205,206*
Comment: Every even x from 38 through 66 is at least almost-prime, with four in a row prime; and high density given parity restriction.
Base 17: x^8+3x^7+8x^6+9x^5+8x^4+5x^3+4x^2+12x+15
1,2*,4*,7,8*,9,12,13,17*,22*,29*,31,33,37*,38,43,44*,52*,59,64,67*,68,73,74,94,100,109*,
115,122*,133*,143,152*,154,158*,163,169,172,173,182,184,193*,194,203,209*,217,219,224,
238*,239,242,247*,263,268,269,273,279,297,303,319,338,340,358,359,364,367,373,379*
Comment: Nothing very remarkable.
Base 18: 13x^7+13x^6+6x^5+9x^4+7x^3+13x^2+15x+7
1*,2,3,4,5,8,9,11,12*,13*,16,18*,20*,23,25*,29*,32*,40*,43,44*,45,46*,48,49,51,54,55*,
58*,62*,64,65,68,69*,71*,75,78,81,82,85*,87,88,90,92,96,97,102,103,104,107,110,112,113*
Comment: Pretty high density of primes and almost-primes.
Base 19: 9x^7+7x^6+15x^5+15x^4+12x^3+3x^2+17x+1
1*,2,3,5,6*,9*,10*,11,13,18,19*,21,24*,25*,29,30,34,37*,42,51,52*,57*,58*,67,70,72,81,87,88*,90*,94*,97,101,102*
Comment: Primes cluster a bit, but otherwise unremarkable.
Base 20: 6x^7+11x^6+9x^5+3x^4+8x^3+7x^2+x+13
1,2,3,6,16,18,20*,31,36,38,46,53,58,60,76,80*,100,106,116,118*
Comment: Quite sparse.
Base 21: 4x^7+14x^6+2x^5+10x^3+13x^2+20x+10
1*,2,3,5,7,9,12,13,21*,23,24,27,29,31*,33,36,38,44,49*,53*,54,55,57*,58,61,63*,67,71*,73*,77,79,81*,87,89,91,94,97*,99,101*
Comment: Pretty unremarkable.
Base 22: 3x^7+8x^6+4x^5+11x^4+4x^3+1
1*,6,12,13,15,16,19*,21*,22*,24,25,27,33,36*,42,49*,55,57,58,61*,66*,72*,75,78,79,90,91,100,111*
Comment: Primes at x=19, 21 and 22 surprising considering the gaps on either side.
Base 23: 2x^7+10x^6+19x^5+3x^4+17x^3+7x+9
1*,2,4,5,6,7*,8,10,13*,14*,16*,17,19,20,21,23*,24,25,26,28,29,37,38,40,43*,
44,46,47,49,52,53,64,65*,67*,70,73,74,76*,77*,85,86,87,88,91,92*,94,97,100*
Comment: Reasonably high density of primes and almost-primes, and a string is followed by a gap.
Base 24: x^7+20x^6+14x^4+12x^3+8x^2+22x+1
6,8,10,14,20,24*,30,33,36,38*,41,44*,50,56*,66,68,76,80,84,86*,89,98,104,108,110,114,120,128,134*
Comment: Nothing remarkable.
Base 25: x^7+9x^6+11x^5+13x^4+4x^3+4x^2+23x+8
1*,2,3,5*,6,7,11,13,15,19,21*,25*,27,30,31,37,39,41*,42,45*,47*,51*,55,63,66,71,75,77*,81,82,87,89,91,93,95,105,111*
Comment: Fairly high density among odd x.
Base 26: x^7+x^6+6x^5+2x^4+23x^3+21x^2+8x+21
1*,2,4,7,8,10,11,17,19,22,25*,26*,31,32*,35,38,41,46,47*,50,52*,54,58,61*,70,71,73*,74,80,85,89*,92,95,96,106,109,110*
Comment: Nothing worthy of mention.
Base 27: 21x^6+19x^5+9x^4+5x^3+8x^2+4x+7
2,4*,5*,6,7,8,9*,11,13,15,16,18,22,25,26,27*,29*,32,33,34,36,39,40,43*,45,48*,
49,52,53*,55*,57*,62,71,72*,75,79,83,85,86,88*,92*,93*,95*,97*,104,105,106,110*
Comment: Primes are clustered in the list to a high degree.
Base 28: 17x^6+12x^5+23x^4+25x^3+x^3+20x+17
1,2,3,4*,5*,6,7*,8,9*,10,12,13,14,15,18,20,23,25,28*,30*,31,33,37*,39,40,41,42,43,44*,
45*,48,49,50,54,55,58*,63,64*,65,66,67*,69,72,75,80,83,89,93,94,95,97,98*,100,104*
Comment: High density, with all x up through 15 except 11 making the list.
Base 29: 14x^6+4x^5+5x^4+9x^3+23x^2+17x+1
1*,2*,3,5,6,9,10,11,12,14*,15*,16*,17*,19,22,23,24,25,28,29*,31,32,34,42,45,46*,54,56,57,
60,61*,62,63,65,67,68,70*,71,72,74,80,81,82*,85,91,92,93,94,95*,96,97,98,100,104,109*
Comment: High density, and a string of primes for x=14 through 17.
Base 30: 11x^6+16x^5+6x^4+25x^3+13x^2+4x+13
2,6,10,14*,20,28,30*,35,36,54,64*,74,75,76,79,80*,90,110,114,124*
Comment: Nothing all that remarkable aside from absences.
Base 31: 9x^6+14x^5+27x^4+2x^3+11x^2+11x+29
1*,2*,3,4,5,6,7,9,10,11*,14,16*,17,19,20,21*,22,23,24,25*,30*,31*,32,35,36*,37*,42,44*,45*,
46,51*,52,56,57,59*,60*,65*,70,74,80,82,84,86,91,92,94*,95,99,101,102,111,112,114,115*
Comment: Pretty high density and a tendency for primes to arise in pairs.
Base 32: 7x^6+26x^5+23x^4+18x^3+31x^2+17
1,2*,4,5,6,8*,9,10,12*,14,15,18*,19,20,24,27,30,32,35,40*,41,46,48,49,52,54,60*,61,64,65,70*,71,76,77,78,79,86,88,90,92,100*
Comment: Nothing very remarkable.
Base 33: 6x^6+16x^5+32x^4+11x^3+25x^2+22x+1
1*,2,3*,4,6,7,12,13,15,16*,21,22,25,27,28*,31,33*,35,36,37,39,42,43,45,46,51,
52,53,57*,58*,60*,61,69,70,75*,76,78,85,88,90*,91,96,97,98,100,101,102,103*
Comment: High density and a string of three primes, where x=59 is simply excluded.
Base 34: 5x^6+15x^5+5x^4+28x^3+9x^2+23x+15
2*,4*,8*,12,14,28,32,34*,38,44,54,58,68*,74*,92,112,140,152,158*
Comment: Nothing that special here.
Base 35: 4x^6+20x^5+6x^4+19x^3+23x^2+8x+3
1*,2,4,5*,8,9,11,13,15,20,23,28,34,35*,38,40,41*,43,44,46,50*,56,58,61,70*,71,73,83*,89*,95,103,104,106*
Comment: Fairly unremarkable.
Base 36: 3x^6+31x^5+5x^4+3x^3+7x^2+34x+25
2*,4,14*,18,24,32*,36*,56*,66,72,74,84,94,98,108,114,128*
Comment: Nothing here.
Base 37: 3x^6+10x^5+12x^4+4x^3+25x^2+34x+9
1*,2,3,4,5*,8*,10,11,15,16,19,20,23,25,26*,29,32,34,35,37*,41*,44,47,49*,52,53,56,59,64*,65*,66,67*,70,74,76,77,83,85,86,87,91,92*,97*,100*
Comment: x=64 through 67 are surprising.
Base 38: 2x^6+30x^5+6x^4+34x^3+29x^2+18x+1
2*,4,8,12,14*,18*,20*,24,28,30,38*,44*,50,62,68,72,74,78,84,86,98,104,110,116,118,126,130,132*
Comment: Very large gap between primes at x=44 and x=132.
Base 39: 2x^6+15x^5+9x^4+28x^3+9x^2+18x+34
1,2,3,5,7*,9*,10,11,14,15,18,23*,29,35,37*,39*,42,44,45,49*,53,57,58,59,60,67,77,79*,83,89*,93,99,105*
Comment: Nothing too special here.
Base 40: 2x^6+2x^5+6x^4+18x^3+21x^2+30x+33
4*,6,10*,12,24,28,40*,42,54,58,70,82*,90,118,124,126,152,154,166,172*
Comment: Again, nothing special.
Base 41: x^6+31x^5+25x^4+15x^3+13x^2+16x+12
1*,3,7,23*,29,37,41*,43,49*,55,77,79,83,85,91,99,107,111,113,115,121*
Comment: Gap between x=49 and x=121.
Base 42: x^6+22x^5+15x^4+32x^3+34x^2+41x+31
2,4,6,7,8*,10*,12,14,15,16,18,24*,26*,28,30,32,35,38*,42*,44*,48,50,58,64,66,72,78,86,92,94,96,102*
Comment: Primes clustered, and a near absence of odd x generating twice primes.
Base 43: x^6+14x^5+9x^4+38x^3+40x^2+26x+29
1*,2*,3*,4,5,6,7*,9,10*,11,12,13,14,15*,16,17,19*,20,23*,24,25,26,27*,29,30,32*,33*,34,36,38*,39,40*,41*,43*,45*,46*,48*,49*,
50*,53,54,56,57*,59*,61*,62*,63,67*,68,69,70,72,74,76*,77*,78,79*,80,81*,82*,83,85*,88*,89*,90,91,95,96,98,100,101*,102*,103*
Comment: Extremely dense, and eight primes from x=40 through x=50.
Base 44: x^6+7x^5+30x^3+33x^2+1
2*,6,8,12*,14*,16,20,22,23,24,26*,30,35,36*,38,42,44*,52,54,59,62,63,66,68,71,74,75,80,84,86,96*,98*,104,106,108,110*
Comment: Nothing all that noteworthy.
Base 45: x^6+26x^4+32x^3+24x^2+7x+43
1,2,3,5,6,7*,11,12*,14,17,22,24,25*,28*,30,32,33,35,38,40,41*,45*,52*,53,56,61,62,63*,67,68,69,73,77,83*,94,96,98,100,101*
Comment: Again, this base seems mundane.
Base 46: 40x^5+39x^4+2x^3+5x^2+38x+9
1,6,10,13,16*,19,21,25,28,34*,40,46*,49,55,61,70,76,84,91,93,94*,100,103,115,124,130,133,136,139*
Comment: Nothing here.
Base 47: 36x^5+32x^4+7x^3+22x^2+30x+20
1*,9,11,23*,33*,39,47*,59,65,69,77,81,87,89,93*,95,103,107,111,119,125,129,147,153*
Comment: Nope.
Base 48: 33x^5+43x^3+26x^2+11x+1
2,6*,10,18,24,28,30*,36*,45,48*,52,60*,72*,78*,84,90,93,98,100,102,105,108,116,120*
Comment: Not all that remarkable.
Base 49: 29x^5+38x^4+21x^3+13x^2+6x+38
1,9,12,13,19,22,24,25*,27*,31,35,39,40,49*,55,63,67,82,87,99,102,105*
Comment: First two primes at x=25 and x=27.
Base 50: 26x^5+46x^4+6x^3+38x^2+36x+33
1,2*,4*,5*,8,9,14,17,19*,22,23,25*,28*,30,32,34,35*,37*,40*,43,47,49*,50*,53*,56,
58,62,64,65*,68,77,79,83,89,95*,98*,103,106,109,113,116,119,124,128,131,134*
Comment: Nothing worth mentioning.
Base 51: 24x^5+19x^4+31x^3+34x^2+26x+49
1,3*,5,9,12,15*,18,21,23,24,25,30*,32,33,36,39,44,51*,52,54,57*,59,60,63,64,66,
71,72,78,81,102,108,111,114,115,117,120,123,124,129,134,136,138,141,142,144,150*
Comment: Twenty-five almost primes between primes at x=57 and x=150.
Base 52: 22x^5+6x^4+35x^3+24x^2+43x+21
1*,2*,3,4*,5*,7,8*,9,10,16*,17,18,19,22,23,25*,26,31,35,36,37,38*,40*,43,44*,50*,52*,
53*,54,55,58,59,61,62,64,65*,67*,68,73,74,75,79*,80,82*,85,86*,88*,89*,93,94*,95*,106*
Comment: High density and notable prime clustering.
Base 53: 20x^5+6x^4+14x^3+3x^2+26x+4
1*,3,5*,7,9,11*,13,15,17,19*,21,23,27*,29*,33,35*,37,43*,45*,49,51,53*,55*,57,59,63,65*,69*,71,79,83,85*,87,91,93,95,99*,101,105,107,109*
Comment: Incredibly dense among the odd numbers.
Base 54: 18x^5+17x^4+24x^3+15x^2+29x+7
2,4*,5,6,8*,10*,18,20,22,24,25,40,44,48*,54*,60,64,66,69,74*,84,85,88*,90,94*,104*
Comment: Nothing that remarkable.
Base 55: 16x^5+39x^4+23x^3+31x^2+37x+23
1,2,3*,4,5*,6,7*,8*,10*,11*,12,13,17*,19*,20*,21,24,26,28,30,31,32,34,38,41*,42,43*,44,45*,47*,49,53,
55*,56,57*,58,60*,63,64*,65,66,68,70*,75*,77,78*,84,85,86,87,88*,89*,90,94,96,97*,98,99,104,106*
Comment: Very dense.
Base 56: 15x^5+15x^4+27x^3+35x^2+24x+17
1,2,3,4*,5*,6*,7*,10,11*,12,14,15,17,18,19*,20,21,25,27,28,31*,32*,34,35*,37,40,41,48*,54,55,56*,60,61,
63,65,66,67,71,73,74,75,77,80,82,83,84,86,87,89*,90,91,93,95,96,97,98,99,101,102,103,104,108,109,116*
Comment: Extremely dense, and with a near absence of primes after x=56.
Base 57: 13x^5+56x^4+x^3+8x^2+50x+1
1,2*,5,6*,8,9,11*,12*,14,15,20,22,24,26*,27,29,30,33,36*,37,39,40,42,44,45,
47*,50,51,57*,59,60,65,68,69,72,74*,75*,77*,83,84,86,90,92,93,95,96,102*
Comment: Pretty dense, and there is a string of three primes.
Base 58: 12x^5+47x^4+26x^3+34x^2+52x+1
2,4*,6*,8,10*,14,18,20,26,30,34*,36,40*,50*,56,58*,64*,68,76,86,90,94*,96,100,106*
Comment: Pretty dense among even x.
Base 59: 11x^5+45x^4+18x^3+57x^2+51x+51
1*,2*,4,5,6,7*,8*,11*,13,14,16,18,19*,22*,25,28*,29,31,32,35*,37,38,40*,41,44,
45,46,47,50,52*,53,59*,61*,63,64,65,67,70,71,74,79*,80*,84,87,97*,98,103*,104*
Comment: Noteworthy prime clustering.
Base 60: 10x^5+49x^4+10x^3+40x^2+47x+13
1,2*,3,5*,7,8*,10,11,12*,15,17,18*,19,20,21,22,23,28,31,32,33,37,38,42,43*,45*,46*,49,
50,51,52,55,56*,57,60*,63,67,68,70*,71,72,75*,77,81*,85,88,93*,94,98*,99,100,102,103*
Comment: Pretty dense.
Base 61: 9x^5+58x^4+39x^3+18x^2+34x+35
1,2,3*,4,6,7,8,9,11,12*,13,16,17*,19,22,23*,24,25,29*,31,32*,33,34*,37,39*,42,44,47*,48,50,51*,
53*,55,58,59,61*,64,66,68,69,72,73,74,75,76,79,82,83*,92,94,95,96,97,99*,101,102,108*,109*
Comment: Very dense.
Base 62: 9x^5+11x^4+23x^3+33x^2+52x+29
1*,2,3,4,5,6,7*,8*,9,11,12,13,14*,15*,16*,17,19*,20*,22*,24,25*,26,27,28,30,31*,33,34,35,36,38,39*,41,42,43,44,45,
46,47,48,49*,50*,53,54,55,56,59,60*,61,62*,66,69,70,71,73*,74,75*,76,80,81,82*,83,84*,86,89,91,93,94,95,96*,97,100*
Comment: Incredibly dense.
Base 63: 8x^5+30x^4+5x^3+3x^2+55x+52
5*,6,9,11*,15,23*,29,35,41,51*,63*,71,74,83,85,89,93*,95,105*
Comment: Meagre.
Base 64: 7x^5+53x^4+30x^3+23x^2+48x+17
1,2,4,6,9,10,12,14,18,20,22*,24*,26*,28,37,40*,42,44,50,52,56,57,58,62,64*,66*,70,78,80,89,94,96*,104*
Comment: Nothing special.
Base 65: 7x^5+16x^4+20x^3+49x^2+45x+8
1,2,3,5,9*,14,15,19,23,25,27,35,37,39,49,53,57*,63,65*,69,74,79*,87*,93,102,115,121,122,123,127*
Comment: x=9 generates only prime before x=57.
Base 66: 6x^5+47x^4+26x^3+14x^2+44x+1
2*,8*,12,14,15,16,18,20,21,26,29,30,33,38*,39,42*,44,46,48,54,59,60,63,64,66*,68,72,74,78,86*,88,89,96*,98,110*
Comment: Pretty dense among even x.
Base 67: 6x^5+15x^4+34x^3+23x^2+2x+53
1,2*,4*,5*,7*,9*,10,11,12,14*,16,17,18,19,23,25,26*,28*,29,31,35,37,38,40*,42,
44*,50,51,56,59,63*,64,67*,70,71,72,74,75,81,84*,86,91*,93,95,99,102,103*
Comment: Pretty dense.
Base 68: 5x^5+53x^4+33x^3+19x^2+28x+49
1,2*,3*,4,5,6*,11*,13*,16,17,18,23,24,25,26*,27,30*,31*,33*,38*,40,44,45,46,48*,50,53,
58*,60,61,68*,71*,72,73*,74,78,81,82*,87,90*,95*,96*,97,102,103,104,106,107,108,115*
Comment Fairly dense, but not remarkable.
Base 69: 5x^5+26x^4+11x^3+48x^2+40x+55
1,2*,4*,7,9,10,12*,13,15,17,18,20,21,24*,26,32*,34,37,39,41,43,47*,52,54*,57,62,
67,69*,70,74,76,77,79*,83,84,85,87,89,91,92,94,97,105,109,114,119,122,127,129*
Comment: Large prime gap from x=79 to x=129.
Base 70: 5x^5+28x^3+49x^2+39x+3
2*,4,6,8,10,12,14,16,20*,22*,26,28*,38,40,42,44,46,48,50,52,58,62,64,68*,70*,76*,78,86,88*,92,96,98*,100*
Comment: Very dense among even x, substantial prime clustering.
Base 71: 4x^5+47x^4+5x^3+57x^2+49x+51
1,2*,4,5*,8,11,14,19,20,26,29*,35,36,37,41,44*,47*,50*,55,56,60,62*,65*,67,
71*,74,77,80*,86,89,90,91,95,99,101,107,110,112,116,122,125,130,146,149*
Comment: Fairly dense, with a gap from x=80 to x=149.
Base 72: 4x^5+25x^4+4x^3+64x^2+71x+25
1*,2*,3*,4,5,7*,8*,11*,12,16,17*,18,22*,23,26,28,30,32*,35,38*,42*,43,44,46*,
51,53,57*,58*,61*,63,66,67*,68,71,72*,77,78,81,82,87*,91,92,93,98*,101*
Comment: Fairly dense, some clustering.
Base 73: 4x^5+4x^4+19x^3+14x^2+21x+35
1*,3,4*,6*,9,12,13,16*,19,21,22,24*,27,31*,33,34,37,39,43,46,48,49,52,64*,66*,73*,76,78*,85,87*,88*,91,96,97,99,102,103*
Comment: Nothing too special here.
Base 74: 3x^5+58x^4+42x^3+15x^2+54x+9
1*,2,4,5,7,13,14,19,20,22,23,25*,26,28,29*,32,34*,35*,37*,40,43,44*,47,49,50,53*,59,61*,62,
64*,68,71,74*,77,79,80,82,83*,85,86,88,91,95,97,98,104,106,110,116,119,124,125,128,131,133*
Comment: Large prime gap from x=83 to x=133; string of primes at x=34, 35 and 37; and square values at x=5 and 16.
Base 75: 3x^5+40x^4+67x^3+56x^2+7x+58
3*,5,11,15,17,27,33,39,45,47,59,63,68,69,75*,77*,81*,89,91,95,105,123,129,137,143,147*
Comment: Between x=3 and x=147, there is only the string of primes at x=75, 77 and 81.
Base 76: 3x^5+24x^4+13x^3+64x^2+28x+1
1,4,6,7*,10,16,18,21*,22,27,30,37,45,46,51,55,60*,70,72*,73,76*,85,90*,97,100,101,105,111*
Comment: Nothing remarkable.
Base 77: 3x^5+8x^4+25x^3+60x^2+12x+45
2*,4,5,7,8*,11*,14,16,17,23*,26,29,31*,38*,41,50,53*,56,59*,62,68*,71,74,76,77*,83*,89,92,95,98,101*
Comment: Nothing remarkable.
Base 78: 2x^5+71x^4+23x^3+2x^2+28x+73
1*,3,4,5,6,10*,12,13,16,18,19*,22,23,24*,25*,27*,33*,36*,37,43*,45,46,48,57,
59,60*,62,63,66,67,69*,70*,73,76*,77,78*,82,85*,87*,88,90,96*,97,99*,100,103*
Comment: Fairly dense, string of primes.
Base 79: 2x^5+58x^4+20x^2+57x+14
1*,3,9,15*,16,29,31*,37,43,45*,48,55,61*,64,75,79*,93,97,99*,103,115*
Comment: Rather meagre, nothing special.
Base 80: 2x^5+45x^4+32x^3+25x^2+35x+33
2,4*,7,8*,10,12,14,16,19,20,23,26,28*,30,32,40,47,52*,56,58,76,80*,86,91,92,94,100,104*
Comment: Nothing here either.
Base 81: 2x^5+33x^4+36x^3+15x^2+73x+34
1*,3*,5,13,15*,19,24,33*,35*,36,40,41,43*,45,49*,53*,55*,59,61,63,69,71*,75*,79,80,81*,83,88,91,95*,97,99,103,111*
Comment: A little prime clustering, fairly dense in odd x.
Base 82: 2x^5+22x^4+7x^3+3x^2+28x+53
1,2*,3*,4,5*,6,8*,9*,13*,17,18,19,20,22*,25*,26,27,28,32*,33*,34,35*,37*,38*,39,40,43*,44,
47,49,50,51,52*,57*,59,60,64*,68*,69,72*,73,77*,79,82*,83*,84*,85*,87*,89,90,95,100,102*
Comment: Very dense overall with a large number of primes including one very long string.
Base 83: 2x^5+11x^4+23x^3+10x^2+48x+41
2*,3*,4,5,8*,9*,14,15,17,18,20,22,23,24*,29*,30,33,35,38,39,42,44,47,48*,50,53,
62,63,65,67,68,69*,72,75,78*,81,83*,84,87,90*,92,93,95*,99*,102,104*,105*,107*
Comment: Fairly dense and ending with string of three primes.
Base 84: 2x^5+82x^3+71x^2+62x+73
2*,4*,6,8,10,12*,14,16,18,20*,22*,25,28,29,32,34,35,36,39,40,42,44,46,48,50*,53,
57,60,64*,68,70*,74,75,76,78*,80,82,84*,85,88,89,90,92,96,98*,102,106,110,112*
Comment: Incredibly dense in even x (and see final note).
Base 85: x^5+76x^4+14x^3+62x^2+5x+83
1*,3*,4,6*,7*,10,13,15*,17,18*,21,24*,31,32,34,35,39*,41,42*,43*,45,46,48*,52,53,55*,57,59,60,63,70,71,72,75,81,85*,87,88,90*,94,95,100,102*
Comment: Pretty dense, given that 2 modulo 3 is largely excluded.
Base 86: x^5+67x^4+69x^3+31x^2+56x+29
1,2,3*,4,5*,6,9*,10,12,13,14,15*,16,17,18*,20,22,24,25*,26,27,28,29,30,31*,34,36,39,40,41*,43*,44,46,49,
52*,54*,57,58,61,62,63*,66,68,69,71,75,76,77*,80,81,82,83,84*,86*,88*,91,92,94,95*,99,101,103,105,107*
Comment: Incredibly dense.
Base 87: x^5+59x^4+74x^3+41x^2+25+1
1,2*,3,4,5,6,7,8,9,10,11*,12,14*,15,16,17,19,20,21,22,25,26,29*,32,34,35,39,40,41,42,47*,48,50*,53,
54*,55,56,59,61,62*,66,69,71,72*,74*,75*,76,77*,78,80,81*,86,87*,89*,90,92,93,95,98,99,101,102*
Comment: Again, very very dense, especially since 1 modulo 3 generates multiples of 3.
Base 88: x^5+52x^4+25x^3+74x^2+22x+1
2,4*,5,6,7,9*,10*,11,13,14,15,17,19,20*,23*,24,25,28*,30,32,33,39,40,45,48,
49,52,53,54,56,59*,63,66,67,70,72,74,75*,79,83,84,88*,90*,95*,96,97,99,100*
Comment: Fairly dense.
Base 89: x^5+45x^4+8x^3+31x^2+13x+79
1,3,4,5,6,8,9,11,14*,18,19,21,23,26*,29,30,33,36,39,44*,45,47,48,50,51,53,54*,56,
63,64,65,66,78,80,84*,86,89*,93*,96,99*,100,105,107,108,114,123,125,126,131*
Comment: Unremarkable.
Base 90: x^5+38x^4+20x^3+84x^2+71x+43
1*,2*,3,4,5*,6,8,9*,11,12,13*,15,16,17,18,19,20,22*,23*,26*,27*,28*,31,32*,34,35*,37*,38*,42,44,45*,46,48*,49,51,52,53,54,
55*,56,60*,61,62*,63,64,65*,66,67*,69,70,71,73*,74*,76*,78,79*,81,82*,83,84*,87*,88,89,90*,91,93*,94,95*,96,97,99,100,101*
Comment: Both enormously dense and with a lot of prime clustering.
Base 91: x^5+31x^4+62x^3+57x^2+89x+73
1*,3*,4,6,7,9*,10,13,15*,18,21,24,25*,28*,31,33,34,36,37,45,46,48*,49*,51,55,57,
58*,64,66,69,70*,72,78,79,82,84*,87,88*,90,91*,93,94,97,105,106,108,111,112*
Comment: Not very remarkable.
Base 92: x^5+25x^4+40x^3+47x^2+42x+9
2*,4,14,16,20,22*,28,32,40,50*,58,64*,68,76,86,88,92*,100*
Comment: Nothing special here.
Base 93: x^5+19x^4+43x^3+66x^2+65x+91
2,3*,5*,8*,9*,11,18,20,23*,24*,27,33*,38,44*,45,50,54,59,60*,68,74,78,80*,87,93*,94,99*,101,102,110,113,120*
Comment: Pretty meagre.
Base 94: x^5+13x^4+71x^3+40x^2+85x+67
1*,2,4,5*,6,9,10*,11,12,13,14,15,18,20,21*,24*,25*,29*,30,31,34*,35,36,39*,
40,44,47,49,50,51,55,61*,64,66,68,69,70,72,76,77,81,88,90,91*,94*,95,100*
Comment: Little cluster of primes.
Base 95: x^5+8x^4+27x^3+86x^2+75x+58
3*,5*,7,17*,23,29,33*,35,39,43,45,47,57,63*,65*,71,72,93*,95*,99*,105,107,113*,115,117*
Comment: Fairly dense among plausible values, string of three primes.
Base 96: x^5+3x^4+5x^3+42x^2+53x+49
2*,3,5,6,8*,12,15,17,21,23,26*,29,33,36*,38,45*,47,48,51,61,65*,66,68*,71,77,78*,80*,81,83,87,93*,95,96*,101*
Comment: Unremarkable.
This listing derives from taking the designation of a particular hootoo article, A41078748, and reading it as a base-13 number. If there really is something remarkable about it, it would have to be said that I was mediating or channelling something to come up with it. This brings me to my final note about the listing. In my investigation of the base-84 case, I went through the process initially with a multiplication sign in place of the exponentiation in the cube term, so that I was really investigating the polynomial 2x^5+71x^2+308x+73. The unbelievable thing is that while the correct polynomial generates primes and almost-primes at about as high a frequency as possible, the falsely substituted polynomial is even more extreme in this direction. Was I responding to some external stimulus in making my error?
Web-based factoring resource: http://www.alpertron.com.ar/ECM.HTM
essentially being that which arise from considering this common evaluation--8413346833 in base ten--in different bases. After each polynomial
is the beginning of the list of values of the variable which generate a prime or an almost-prime (product of two primes), with those generating
primes marked by an asterisk. The bases that generate each polynomial are shown for convenience.
Base 2: x^32+x^31+x^30+x^29+x^28+x^26+x^24+x^22+x^21+x^20+x^19+x^16+x^14+x^13+x^12+x^11+x^10+x^4+1
1*,2*,7*,16,29,30,31,32,33,36,37*,43,51,64,70,72,80,100,103*,120,148*,150,170,192*,201,204*,224,241*,
246,247,254*,260,261,269,271,275,292*,311,315,318,344,346,352,359,365*,379,380,386,432,436,444,
448,453,456,464,470,483,486,505,518,530,531,534,546,551,556,563,565,574,591*,593*,622,651,658,
681*,721*,729,733,752*,768,771,781,787,793,794,796,805,828,843,866,868*,870,871,878,885,887*,897*,
904*,906,908,923,924*,927,939,950,973,982,983*,988,990,1003,1071,1094,1101,1115,1163,1164,1167*,
1181,1197,1216,1221,1232,1242,1270*,1279,1281*,1317,1318,1323,1326,1331,1335*,1360*,1372,1417,
1421,1456,1505,1506,1517,1519,1531*,1538,1558,1563,1571,1584*,1589*,1594,1611,1626,1641,1652,1699,
1703,1767,1772,1812,1813,1843,1848,1858,1863,1875,1885,1912,1927,1943,1951,1970,1972,1988,2009,
2018,2030*,2051,2060*,2067,2086,2087*,2089,2105,2115,2121,2122,2136*,2140,2143,2150,2165,2177,2182,
2191,2193,2250,2254,2255,2261,2268,2316,2317,2325*,2332,2362,
Comment: Only four primes or almost-primes through 28, then five consecutive; -1 only prime generator among negative x until -84.
Base 3: 2x^20+x^19+2x^17+x^15+x^14+x^10+2x^9+2x^7+2x^6+x^4+x^3+2x+1
1*,3*,4,7*,10*,12*,13*,15*,22,31,36*,37,41,42,48,52,54*,60,61,64,69,73,75,76,79,88*,89,
90*,96,99,111*,113,121*,129,132,135,139*,147,148,151,159,163,169*,178,184*,205,207,225*,
230,235,257,258*,262*,265,268,274,285*,289,291,297,304,334,340,359,363,372*,375,381,382*
Comment: As many primes as possible from 10 through 15, and primes at 88 and 90 with 3 times a prime at 89.
Base 4: x^16+3x^15+3x^14+x^13+x^12+x^11+3x^10+2x^9+x^8+x^7+3x^6+3x^5+x^2+1
1,3,4*,7,10*,15*,25*,28*,33,40,42,45,50,58,67,69,72,75*,78*,79,82*,85,90,93,102*,103,106,
108,109,114,117,118,120,127,132,142,145,148,163,168,169,177,208,219*,220*,222*,234*,242,
246,247,262,278,283*,285,288,295,309,325,330,332,336,345,349,358,369,372,375,384*
Comment: x=51 yields 5^9 times a prime, and, after a gap beginning with a prime at x=102, there are primes at x=219, 220 and 222.
Base 5: x^14+x^13+4x^12+2x^11+x^10+2x^9+3x^8+4x^6+4x^4+4x^3+3x^2+x+3
1,2,5*,8*,16,17,27,29*,35*,41,44,50,63,66,70,80,83,98,125*,128,131*,134*,158,170,176,
178,187,190,200,203,205,206,216,223,248,260*,275,290*,299,302*,308,314*,329,341*,344,
347,353,360,380,383,387,395,398,406,431,440,444,454,461,464,486,491,500,512,531,533*
Comment: Primes at 125, 131 and 134, with prior back at 35 and next not until 260; big gap 341-533.
Base 6: 3x^12+5x^11+x^10+5x^8+3x^6+x^5+x^4+5x^3+4x^2+4x+1
1,4,6*,15,18*,27,39,42,48*,60*,62,63,68,75,84,96,99,104,106,109,126,130,134,151,152,162,174,180,193,
201,208,231,238,246*,255,274,285,286,291*,294,305,309,330,333,334,348*,351,357,378,393,406,411,412,
427,442,456,465,468,473,483,504,516,531,550,554,561,564,567,568,582,627,636,642,643,663,678,690*
Comment: Primes rare, even considering the necessary condition that 3 divides x.
Base 7: 4x^11+x^10+5x^9+3x^8+3x^7+x^5+6x^4+6x^2+5x+3
1*,2*,5,7*,8,10,11*,16*,17,19,25,28,34,36,40*,44*,47*,50,55,56*,59,62,67,71,73*,80,81,82*,86*,90,95,
99,101,105,106,113,122,125*,127,154,157*,158*,170,176,185,187,189,191,193*,197,200,205,207,208,209,
223,224,227,233,235,241,242*,245*,248,249,254,260,265,268,269,272,275,276,278,288,295,296,311*,319,
320,325,334,335*,338,340,341,343,347,359,362*,364*,365,368,374,387,391*,394,396,397*
Comment: Some clustering of primes, with almost-primes also seeming to come in swarms.
Base 8: 7x^10+6x^9+5x^8+3x^7+6x^6+2x^5+7x^4+6x^3+2x+1
2*,8*,12,14,15*,18,23*,24*,29,33,42*,45,47*,53*,54*,57*,65*,68,74,77,78,80*,81,87,92,98,
107*,108*,113,117,119,120,128*,143,144*,147*,152,158*,161,165*,168,183*,185,203,206,207,
209,219*,227*,239,240*,242*,243,252,257*,260,264,284,287,308*,312,315,318,330,339,341,345,
362,378,383,392*,395*,399,407,410,414*,419,420*,425,438*,443,449,450,452,453*,455,465,470*
Comment: Early large cluster of primes, with clustering of both primes and almost primes continuing.
Base 9: 2x^10+3x^9+6x^8+4x^7+x^5+6x^4+8x^3+x^2+3x+7
1*,3*,4,5,6,9*,10,12*,15,16,19*,22*,25,26,27,30,31,33,34*,36,37,39*,40,45,46,58*,66,75,76,78,
81,82,85,94,99,103,109,120,129,139,142*,144,148,152,160,166,180,186,190,193*,195,214,216,226,
237*,250*,256*,267,274,276*,278,282,283,292,295,304,306,319,323,324,325*,337,339*,346,349,358,
367*,373,408,414,423,427,430*,433,438,439,447,448,457*,459,460,462,463,466,471,480,484,492,505*
Comment: String of almost-primes with just one prime, slight evidence of clustering.
Base 10: 8x^9+4x^8+x^7+3x^6+3x^5+4x^4+6x^3+8x^2+3x+3
1*,2*,3,4*,8*,9,10*,11,14,16*,22*,23,28,31,35*,41,46,47,49,59,61,62*,64*,68,70,73,76*,78,79,83,
85*,92*,94*,99,104*,106,107,108,118*,126,137,139,140,148,151,155,157,160,164,170,171,176*,177,
179,184,190,195,196*,197,199,200,205,206,208,218*,221*,223*,224,225,238*,241,244,247*,248,257,
265,268*,275,277,284,295,299,307,314,316,319,332,338*,344,349,350*,354,364*,368,374,377,380,382*
Comment: Nothing remarkable aside from a little clustering of primes.
Base 11: 3x^9+6x^8+2x^7+8x^6+x^5+2x^4+10x^3+1
1,2,3*,4,9,11*,15,17,20,21,28,29,30*,33,36,39,43,44,48,50,53,59,63*,66*,68*,69,80,90,99,101*,103,
104,105,110,114,119,124,129,134,140,146*,149,153,156,158*,159,164,168,169,174,184,185*,198*,204,
206*,207,213,220,224,228,230,231,233,236,251*,263,264*,267,269*,270,273,281,288,290,294*,299,
305*,306,308,318*,321,323,324*,325,326,339,345,357,358,360,362,365,374,378,379,381,383,384*
Comment: Some clustering, but not very notable.
Base 12: x^9+7x^8+6x^7+9x^6+7x^5+4x^4+2x^3+11x^2+8x+1
2*,4,6,10,12*,14*,16,20*,24*,26,28,32*,38*,42,44,46*,58,60*,66*,68*,70*,72*,86,94*,96,98*,102,104,
110,116,118*,122,138,160,164,168*,174,182,190,196,208,210,214,222,228*,230,240,242,248,252*,256*,
262,264*,266*,268,270,272,274,276,278*,284,290,296,298*,306,308,312,322,336,346,350,356*,366,384*
Comment: Four consecutive primes (66-72) and nine consecutive primes or almost-primes (262-278), odd numbers not considered.
Base 13: 10x^8+4x^7+x^6+7x^4+8x^3+7x^2+4x+8
1,3,9,11,13*,15,17,21,49*,51*,53*,55,57,65,69,75*,81,91*,93,95*,97,103,115*,117,119,129,133*,
137*,139*,143*,145,157,167,171,173*,181,187,189,191,213*,215*,217,219,229,231*,235,237,249,
255,259,273,275,283*,285,289,297,299,307,317*,321*,325,329,335*,341*,343*,347,353,363,371,375*
Comment: Substantial gaps and strings, and some clustering of primes.
Base 14: 5x^8+9x^7+11x^6+5x^5+4x^4+4x^3+6x^2+13x+3
10,14*,26,38,44*,64,74*,80,104*,164*,194,200,212,254,264,304,356,360,470,494,504,506,520,524*,614,660,694,710,
818,840,884*,930,944*,980,1010,1034,1064,1184,1240,1304,1352,1448,1454,1520,1538,1550,1560,1574,1594,1630,1634,
1700,1724,1790,1814,1820,1844,1890,1934,1954,1970,1974,1988,1994,2000,2012,2024,2120*,2135,2150*,...,2330*,...,2900*
Comment: Necessarily very sparse, and, after an enormous gap in primes, there are two primes around the first odd almost-prime generator.
Base 15: 3x^8+4x^7+3x^6+9x^5+4x^4+8x^3+7x^2+8x+13
1*,2,3,4,5,6*,7,8,9*,10*,11,14,15*,16*,19,21*,23,30,31,34*,40*,42,49,50*,51,54,55,56,61,69,71*,76*,
79,85,86*,89,96*,100*,101,103,111,114*,115*,121,124*,126,127,129,132,134*,135,136,139,140,142,
154,157,164*,167,170*,177,180*,181,185*,191,195,196,199,201,209,215*,222,224,226,229,236,241*
Comment: High density of primes and almost-primes through x=23, consecutive primes at 114-5.
Base 16: x^8+15x^7+5x^6+7x^5+9x^4+7x^3+12x^2+x+1
1,2*,3,4,5,6,9,10,13,14*,16*,20,21,23,25,28,29,30,31,32*,38*,40*,42,44,46*,48,49,50,52*,54*,56*,57,58*,60,
62,64*,66,67,72*,73,76,80*,86*,92*,94*,96*,98,100,103,104*,106,113,114*,118,120,122,123,124,126,128,132*,
136,143,144,148,150,152,162,164*,166*,168,171,173,174*,178,179,180,184,188,192,200*,201,202,205,206*
Comment: Every even x from 38 through 66 is at least almost-prime, with four in a row prime; and high density given parity restriction.
Base 17: x^8+3x^7+8x^6+9x^5+8x^4+5x^3+4x^2+12x+15
1,2*,4*,7,8*,9,12,13,17*,22*,29*,31,33,37*,38,43,44*,52*,59,64,67*,68,73,74,94,100,109*,
115,122*,133*,143,152*,154,158*,163,169,172,173,182,184,193*,194,203,209*,217,219,224,
238*,239,242,247*,263,268,269,273,279,297,303,319,338,340,358,359,364,367,373,379*
Comment: Nothing very remarkable.
Base 18: 13x^7+13x^6+6x^5+9x^4+7x^3+13x^2+15x+7
1*,2,3,4,5,8,9,11,12*,13*,16,18*,20*,23,25*,29*,32*,40*,43,44*,45,46*,48,49,51,54,55*,
58*,62*,64,65,68,69*,71*,75,78,81,82,85*,87,88,90,92,96,97,102,103,104,107,110,112,113*
Comment: Pretty high density of primes and almost-primes.
Base 19: 9x^7+7x^6+15x^5+15x^4+12x^3+3x^2+17x+1
1*,2,3,5,6*,9*,10*,11,13,18,19*,21,24*,25*,29,30,34,37*,42,51,52*,57*,58*,67,70,72,81,87,88*,90*,94*,97,101,102*
Comment: Primes cluster a bit, but otherwise unremarkable.
Base 20: 6x^7+11x^6+9x^5+3x^4+8x^3+7x^2+x+13
1,2,3,6,16,18,20*,31,36,38,46,53,58,60,76,80*,100,106,116,118*
Comment: Quite sparse.
Base 21: 4x^7+14x^6+2x^5+10x^3+13x^2+20x+10
1*,2,3,5,7,9,12,13,21*,23,24,27,29,31*,33,36,38,44,49*,53*,54,55,57*,58,61,63*,67,71*,73*,77,79,81*,87,89,91,94,97*,99,101*
Comment: Pretty unremarkable.
Base 22: 3x^7+8x^6+4x^5+11x^4+4x^3+1
1*,6,12,13,15,16,19*,21*,22*,24,25,27,33,36*,42,49*,55,57,58,61*,66*,72*,75,78,79,90,91,100,111*
Comment: Primes at x=19, 21 and 22 surprising considering the gaps on either side.
Base 23: 2x^7+10x^6+19x^5+3x^4+17x^3+7x+9
1*,2,4,5,6,7*,8,10,13*,14*,16*,17,19,20,21,23*,24,25,26,28,29,37,38,40,43*,
44,46,47,49,52,53,64,65*,67*,70,73,74,76*,77*,85,86,87,88,91,92*,94,97,100*
Comment: Reasonably high density of primes and almost-primes, and a string is followed by a gap.
Base 24: x^7+20x^6+14x^4+12x^3+8x^2+22x+1
6,8,10,14,20,24*,30,33,36,38*,41,44*,50,56*,66,68,76,80,84,86*,89,98,104,108,110,114,120,128,134*
Comment: Nothing remarkable.
Base 25: x^7+9x^6+11x^5+13x^4+4x^3+4x^2+23x+8
1*,2,3,5*,6,7,11,13,15,19,21*,25*,27,30,31,37,39,41*,42,45*,47*,51*,55,63,66,71,75,77*,81,82,87,89,91,93,95,105,111*
Comment: Fairly high density among odd x.
Base 26: x^7+x^6+6x^5+2x^4+23x^3+21x^2+8x+21
1*,2,4,7,8,10,11,17,19,22,25*,26*,31,32*,35,38,41,46,47*,50,52*,54,58,61*,70,71,73*,74,80,85,89*,92,95,96,106,109,110*
Comment: Nothing worthy of mention.
Base 27: 21x^6+19x^5+9x^4+5x^3+8x^2+4x+7
2,4*,5*,6,7,8,9*,11,13,15,16,18,22,25,26,27*,29*,32,33,34,36,39,40,43*,45,48*,
49,52,53*,55*,57*,62,71,72*,75,79,83,85,86,88*,92*,93*,95*,97*,104,105,106,110*
Comment: Primes are clustered in the list to a high degree.
Base 28: 17x^6+12x^5+23x^4+25x^3+x^3+20x+17
1,2,3,4*,5*,6,7*,8,9*,10,12,13,14,15,18,20,23,25,28*,30*,31,33,37*,39,40,41,42,43,44*,
45*,48,49,50,54,55,58*,63,64*,65,66,67*,69,72,75,80,83,89,93,94,95,97,98*,100,104*
Comment: High density, with all x up through 15 except 11 making the list.
Base 29: 14x^6+4x^5+5x^4+9x^3+23x^2+17x+1
1*,2*,3,5,6,9,10,11,12,14*,15*,16*,17*,19,22,23,24,25,28,29*,31,32,34,42,45,46*,54,56,57,
60,61*,62,63,65,67,68,70*,71,72,74,80,81,82*,85,91,92,93,94,95*,96,97,98,100,104,109*
Comment: High density, and a string of primes for x=14 through 17.
Base 30: 11x^6+16x^5+6x^4+25x^3+13x^2+4x+13
2,6,10,14*,20,28,30*,35,36,54,64*,74,75,76,79,80*,90,110,114,124*
Comment: Nothing all that remarkable aside from absences.
Base 31: 9x^6+14x^5+27x^4+2x^3+11x^2+11x+29
1*,2*,3,4,5,6,7,9,10,11*,14,16*,17,19,20,21*,22,23,24,25*,30*,31*,32,35,36*,37*,42,44*,45*,
46,51*,52,56,57,59*,60*,65*,70,74,80,82,84,86,91,92,94*,95,99,101,102,111,112,114,115*
Comment: Pretty high density and a tendency for primes to arise in pairs.
Base 32: 7x^6+26x^5+23x^4+18x^3+31x^2+17
1,2*,4,5,6,8*,9,10,12*,14,15,18*,19,20,24,27,30,32,35,40*,41,46,48,49,52,54,60*,61,64,65,70*,71,76,77,78,79,86,88,90,92,100*
Comment: Nothing very remarkable.
Base 33: 6x^6+16x^5+32x^4+11x^3+25x^2+22x+1
1*,2,3*,4,6,7,12,13,15,16*,21,22,25,27,28*,31,33*,35,36,37,39,42,43,45,46,51,
52,53,57*,58*,60*,61,69,70,75*,76,78,85,88,90*,91,96,97,98,100,101,102,103*
Comment: High density and a string of three primes, where x=59 is simply excluded.
Base 34: 5x^6+15x^5+5x^4+28x^3+9x^2+23x+15
2*,4*,8*,12,14,28,32,34*,38,44,54,58,68*,74*,92,112,140,152,158*
Comment: Nothing that special here.
Base 35: 4x^6+20x^5+6x^4+19x^3+23x^2+8x+3
1*,2,4,5*,8,9,11,13,15,20,23,28,34,35*,38,40,41*,43,44,46,50*,56,58,61,70*,71,73,83*,89*,95,103,104,106*
Comment: Fairly unremarkable.
Base 36: 3x^6+31x^5+5x^4+3x^3+7x^2+34x+25
2*,4,14*,18,24,32*,36*,56*,66,72,74,84,94,98,108,114,128*
Comment: Nothing here.
Base 37: 3x^6+10x^5+12x^4+4x^3+25x^2+34x+9
1*,2,3,4,5*,8*,10,11,15,16,19,20,23,25,26*,29,32,34,35,37*,41*,44,47,49*,52,53,56,59,64*,65*,66,67*,70,74,76,77,83,85,86,87,91,92*,97*,100*
Comment: x=64 through 67 are surprising.
Base 38: 2x^6+30x^5+6x^4+34x^3+29x^2+18x+1
2*,4,8,12,14*,18*,20*,24,28,30,38*,44*,50,62,68,72,74,78,84,86,98,104,110,116,118,126,130,132*
Comment: Very large gap between primes at x=44 and x=132.
Base 39: 2x^6+15x^5+9x^4+28x^3+9x^2+18x+34
1,2,3,5,7*,9*,10,11,14,15,18,23*,29,35,37*,39*,42,44,45,49*,53,57,58,59,60,67,77,79*,83,89*,93,99,105*
Comment: Nothing too special here.
Base 40: 2x^6+2x^5+6x^4+18x^3+21x^2+30x+33
4*,6,10*,12,24,28,40*,42,54,58,70,82*,90,118,124,126,152,154,166,172*
Comment: Again, nothing special.
Base 41: x^6+31x^5+25x^4+15x^3+13x^2+16x+12
1*,3,7,23*,29,37,41*,43,49*,55,77,79,83,85,91,99,107,111,113,115,121*
Comment: Gap between x=49 and x=121.
Base 42: x^6+22x^5+15x^4+32x^3+34x^2+41x+31
2,4,6,7,8*,10*,12,14,15,16,18,24*,26*,28,30,32,35,38*,42*,44*,48,50,58,64,66,72,78,86,92,94,96,102*
Comment: Primes clustered, and a near absence of odd x generating twice primes.
Base 43: x^6+14x^5+9x^4+38x^3+40x^2+26x+29
1*,2*,3*,4,5,6,7*,9,10*,11,12,13,14,15*,16,17,19*,20,23*,24,25,26,27*,29,30,32*,33*,34,36,38*,39,40*,41*,43*,45*,46*,48*,49*,
50*,53,54,56,57*,59*,61*,62*,63,67*,68,69,70,72,74,76*,77*,78,79*,80,81*,82*,83,85*,88*,89*,90,91,95,96,98,100,101*,102*,103*
Comment: Extremely dense, and eight primes from x=40 through x=50.
Base 44: x^6+7x^5+30x^3+33x^2+1
2*,6,8,12*,14*,16,20,22,23,24,26*,30,35,36*,38,42,44*,52,54,59,62,63,66,68,71,74,75,80,84,86,96*,98*,104,106,108,110*
Comment: Nothing all that noteworthy.
Base 45: x^6+26x^4+32x^3+24x^2+7x+43
1,2,3,5,6,7*,11,12*,14,17,22,24,25*,28*,30,32,33,35,38,40,41*,45*,52*,53,56,61,62,63*,67,68,69,73,77,83*,94,96,98,100,101*
Comment: Again, this base seems mundane.
Base 46: 40x^5+39x^4+2x^3+5x^2+38x+9
1,6,10,13,16*,19,21,25,28,34*,40,46*,49,55,61,70,76,84,91,93,94*,100,103,115,124,130,133,136,139*
Comment: Nothing here.
Base 47: 36x^5+32x^4+7x^3+22x^2+30x+20
1*,9,11,23*,33*,39,47*,59,65,69,77,81,87,89,93*,95,103,107,111,119,125,129,147,153*
Comment: Nope.
Base 48: 33x^5+43x^3+26x^2+11x+1
2,6*,10,18,24,28,30*,36*,45,48*,52,60*,72*,78*,84,90,93,98,100,102,105,108,116,120*
Comment: Not all that remarkable.
Base 49: 29x^5+38x^4+21x^3+13x^2+6x+38
1,9,12,13,19,22,24,25*,27*,31,35,39,40,49*,55,63,67,82,87,99,102,105*
Comment: First two primes at x=25 and x=27.
Base 50: 26x^5+46x^4+6x^3+38x^2+36x+33
1,2*,4*,5*,8,9,14,17,19*,22,23,25*,28*,30,32,34,35*,37*,40*,43,47,49*,50*,53*,56,
58,62,64,65*,68,77,79,83,89,95*,98*,103,106,109,113,116,119,124,128,131,134*
Comment: Nothing worth mentioning.
Base 51: 24x^5+19x^4+31x^3+34x^2+26x+49
1,3*,5,9,12,15*,18,21,23,24,25,30*,32,33,36,39,44,51*,52,54,57*,59,60,63,64,66,
71,72,78,81,102,108,111,114,115,117,120,123,124,129,134,136,138,141,142,144,150*
Comment: Twenty-five almost primes between primes at x=57 and x=150.
Base 52: 22x^5+6x^4+35x^3+24x^2+43x+21
1*,2*,3,4*,5*,7,8*,9,10,16*,17,18,19,22,23,25*,26,31,35,36,37,38*,40*,43,44*,50*,52*,
53*,54,55,58,59,61,62,64,65*,67*,68,73,74,75,79*,80,82*,85,86*,88*,89*,93,94*,95*,106*
Comment: High density and notable prime clustering.
Base 53: 20x^5+6x^4+14x^3+3x^2+26x+4
1*,3,5*,7,9,11*,13,15,17,19*,21,23,27*,29*,33,35*,37,43*,45*,49,51,53*,55*,57,59,63,65*,69*,71,79,83,85*,87,91,93,95,99*,101,105,107,109*
Comment: Incredibly dense among the odd numbers.
Base 54: 18x^5+17x^4+24x^3+15x^2+29x+7
2,4*,5,6,8*,10*,18,20,22,24,25,40,44,48*,54*,60,64,66,69,74*,84,85,88*,90,94*,104*
Comment: Nothing that remarkable.
Base 55: 16x^5+39x^4+23x^3+31x^2+37x+23
1,2,3*,4,5*,6,7*,8*,10*,11*,12,13,17*,19*,20*,21,24,26,28,30,31,32,34,38,41*,42,43*,44,45*,47*,49,53,
55*,56,57*,58,60*,63,64*,65,66,68,70*,75*,77,78*,84,85,86,87,88*,89*,90,94,96,97*,98,99,104,106*
Comment: Very dense.
Base 56: 15x^5+15x^4+27x^3+35x^2+24x+17
1,2,3,4*,5*,6*,7*,10,11*,12,14,15,17,18,19*,20,21,25,27,28,31*,32*,34,35*,37,40,41,48*,54,55,56*,60,61,
63,65,66,67,71,73,74,75,77,80,82,83,84,86,87,89*,90,91,93,95,96,97,98,99,101,102,103,104,108,109,116*
Comment: Extremely dense, and with a near absence of primes after x=56.
Base 57: 13x^5+56x^4+x^3+8x^2+50x+1
1,2*,5,6*,8,9,11*,12*,14,15,20,22,24,26*,27,29,30,33,36*,37,39,40,42,44,45,
47*,50,51,57*,59,60,65,68,69,72,74*,75*,77*,83,84,86,90,92,93,95,96,102*
Comment: Pretty dense, and there is a string of three primes.
Base 58: 12x^5+47x^4+26x^3+34x^2+52x+1
2,4*,6*,8,10*,14,18,20,26,30,34*,36,40*,50*,56,58*,64*,68,76,86,90,94*,96,100,106*
Comment: Pretty dense among even x.
Base 59: 11x^5+45x^4+18x^3+57x^2+51x+51
1*,2*,4,5,6,7*,8*,11*,13,14,16,18,19*,22*,25,28*,29,31,32,35*,37,38,40*,41,44,
45,46,47,50,52*,53,59*,61*,63,64,65,67,70,71,74,79*,80*,84,87,97*,98,103*,104*
Comment: Noteworthy prime clustering.
Base 60: 10x^5+49x^4+10x^3+40x^2+47x+13
1,2*,3,5*,7,8*,10,11,12*,15,17,18*,19,20,21,22,23,28,31,32,33,37,38,42,43*,45*,46*,49,
50,51,52,55,56*,57,60*,63,67,68,70*,71,72,75*,77,81*,85,88,93*,94,98*,99,100,102,103*
Comment: Pretty dense.
Base 61: 9x^5+58x^4+39x^3+18x^2+34x+35
1,2,3*,4,6,7,8,9,11,12*,13,16,17*,19,22,23*,24,25,29*,31,32*,33,34*,37,39*,42,44,47*,48,50,51*,
53*,55,58,59,61*,64,66,68,69,72,73,74,75,76,79,82,83*,92,94,95,96,97,99*,101,102,108*,109*
Comment: Very dense.
Base 62: 9x^5+11x^4+23x^3+33x^2+52x+29
1*,2,3,4,5,6,7*,8*,9,11,12,13,14*,15*,16*,17,19*,20*,22*,24,25*,26,27,28,30,31*,33,34,35,36,38,39*,41,42,43,44,45,
46,47,48,49*,50*,53,54,55,56,59,60*,61,62*,66,69,70,71,73*,74,75*,76,80,81,82*,83,84*,86,89,91,93,94,95,96*,97,100*
Comment: Incredibly dense.
Base 63: 8x^5+30x^4+5x^3+3x^2+55x+52
5*,6,9,11*,15,23*,29,35,41,51*,63*,71,74,83,85,89,93*,95,105*
Comment: Meagre.
Base 64: 7x^5+53x^4+30x^3+23x^2+48x+17
1,2,4,6,9,10,12,14,18,20,22*,24*,26*,28,37,40*,42,44,50,52,56,57,58,62,64*,66*,70,78,80,89,94,96*,104*
Comment: Nothing special.
Base 65: 7x^5+16x^4+20x^3+49x^2+45x+8
1,2,3,5,9*,14,15,19,23,25,27,35,37,39,49,53,57*,63,65*,69,74,79*,87*,93,102,115,121,122,123,127*
Comment: x=9 generates only prime before x=57.
Base 66: 6x^5+47x^4+26x^3+14x^2+44x+1
2*,8*,12,14,15,16,18,20,21,26,29,30,33,38*,39,42*,44,46,48,54,59,60,63,64,66*,68,72,74,78,86*,88,89,96*,98,110*
Comment: Pretty dense among even x.
Base 67: 6x^5+15x^4+34x^3+23x^2+2x+53
1,2*,4*,5*,7*,9*,10,11,12,14*,16,17,18,19,23,25,26*,28*,29,31,35,37,38,40*,42,
44*,50,51,56,59,63*,64,67*,70,71,72,74,75,81,84*,86,91*,93,95,99,102,103*
Comment: Pretty dense.
Base 68: 5x^5+53x^4+33x^3+19x^2+28x+49
1,2*,3*,4,5,6*,11*,13*,16,17,18,23,24,25,26*,27,30*,31*,33*,38*,40,44,45,46,48*,50,53,
58*,60,61,68*,71*,72,73*,74,78,81,82*,87,90*,95*,96*,97,102,103,104,106,107,108,115*
Comment Fairly dense, but not remarkable.
Base 69: 5x^5+26x^4+11x^3+48x^2+40x+55
1,2*,4*,7,9,10,12*,13,15,17,18,20,21,24*,26,32*,34,37,39,41,43,47*,52,54*,57,62,
67,69*,70,74,76,77,79*,83,84,85,87,89,91,92,94,97,105,109,114,119,122,127,129*
Comment: Large prime gap from x=79 to x=129.
Base 70: 5x^5+28x^3+49x^2+39x+3
2*,4,6,8,10,12,14,16,20*,22*,26,28*,38,40,42,44,46,48,50,52,58,62,64,68*,70*,76*,78,86,88*,92,96,98*,100*
Comment: Very dense among even x, substantial prime clustering.
Base 71: 4x^5+47x^4+5x^3+57x^2+49x+51
1,2*,4,5*,8,11,14,19,20,26,29*,35,36,37,41,44*,47*,50*,55,56,60,62*,65*,67,
71*,74,77,80*,86,89,90,91,95,99,101,107,110,112,116,122,125,130,146,149*
Comment: Fairly dense, with a gap from x=80 to x=149.
Base 72: 4x^5+25x^4+4x^3+64x^2+71x+25
1*,2*,3*,4,5,7*,8*,11*,12,16,17*,18,22*,23,26,28,30,32*,35,38*,42*,43,44,46*,
51,53,57*,58*,61*,63,66,67*,68,71,72*,77,78,81,82,87*,91,92,93,98*,101*
Comment: Fairly dense, some clustering.
Base 73: 4x^5+4x^4+19x^3+14x^2+21x+35
1*,3,4*,6*,9,12,13,16*,19,21,22,24*,27,31*,33,34,37,39,43,46,48,49,52,64*,66*,73*,76,78*,85,87*,88*,91,96,97,99,102,103*
Comment: Nothing too special here.
Base 74: 3x^5+58x^4+42x^3+15x^2+54x+9
1*,2,4,5,7,13,14,19,20,22,23,25*,26,28,29*,32,34*,35*,37*,40,43,44*,47,49,50,53*,59,61*,62,
64*,68,71,74*,77,79,80,82,83*,85,86,88,91,95,97,98,104,106,110,116,119,124,125,128,131,133*
Comment: Large prime gap from x=83 to x=133; string of primes at x=34, 35 and 37; and square values at x=5 and 16.
Base 75: 3x^5+40x^4+67x^3+56x^2+7x+58
3*,5,11,15,17,27,33,39,45,47,59,63,68,69,75*,77*,81*,89,91,95,105,123,129,137,143,147*
Comment: Between x=3 and x=147, there is only the string of primes at x=75, 77 and 81.
Base 76: 3x^5+24x^4+13x^3+64x^2+28x+1
1,4,6,7*,10,16,18,21*,22,27,30,37,45,46,51,55,60*,70,72*,73,76*,85,90*,97,100,101,105,111*
Comment: Nothing remarkable.
Base 77: 3x^5+8x^4+25x^3+60x^2+12x+45
2*,4,5,7,8*,11*,14,16,17,23*,26,29,31*,38*,41,50,53*,56,59*,62,68*,71,74,76,77*,83*,89,92,95,98,101*
Comment: Nothing remarkable.
Base 78: 2x^5+71x^4+23x^3+2x^2+28x+73
1*,3,4,5,6,10*,12,13,16,18,19*,22,23,24*,25*,27*,33*,36*,37,43*,45,46,48,57,
59,60*,62,63,66,67,69*,70*,73,76*,77,78*,82,85*,87*,88,90,96*,97,99*,100,103*
Comment: Fairly dense, string of primes.
Base 79: 2x^5+58x^4+20x^2+57x+14
1*,3,9,15*,16,29,31*,37,43,45*,48,55,61*,64,75,79*,93,97,99*,103,115*
Comment: Rather meagre, nothing special.
Base 80: 2x^5+45x^4+32x^3+25x^2+35x+33
2,4*,7,8*,10,12,14,16,19,20,23,26,28*,30,32,40,47,52*,56,58,76,80*,86,91,92,94,100,104*
Comment: Nothing here either.
Base 81: 2x^5+33x^4+36x^3+15x^2+73x+34
1*,3*,5,13,15*,19,24,33*,35*,36,40,41,43*,45,49*,53*,55*,59,61,63,69,71*,75*,79,80,81*,83,88,91,95*,97,99,103,111*
Comment: A little prime clustering, fairly dense in odd x.
Base 82: 2x^5+22x^4+7x^3+3x^2+28x+53
1,2*,3*,4,5*,6,8*,9*,13*,17,18,19,20,22*,25*,26,27,28,32*,33*,34,35*,37*,38*,39,40,43*,44,
47,49,50,51,52*,57*,59,60,64*,68*,69,72*,73,77*,79,82*,83*,84*,85*,87*,89,90,95,100,102*
Comment: Very dense overall with a large number of primes including one very long string.
Base 83: 2x^5+11x^4+23x^3+10x^2+48x+41
2*,3*,4,5,8*,9*,14,15,17,18,20,22,23,24*,29*,30,33,35,38,39,42,44,47,48*,50,53,
62,63,65,67,68,69*,72,75,78*,81,83*,84,87,90*,92,93,95*,99*,102,104*,105*,107*
Comment: Fairly dense and ending with string of three primes.
Base 84: 2x^5+82x^3+71x^2+62x+73
2*,4*,6,8,10,12*,14,16,18,20*,22*,25,28,29,32,34,35,36,39,40,42,44,46,48,50*,53,
57,60,64*,68,70*,74,75,76,78*,80,82,84*,85,88,89,90,92,96,98*,102,106,110,112*
Comment: Incredibly dense in even x (and see final note).
Base 85: x^5+76x^4+14x^3+62x^2+5x+83
1*,3*,4,6*,7*,10,13,15*,17,18*,21,24*,31,32,34,35,39*,41,42*,43*,45,46,48*,52,53,55*,57,59,60,63,70,71,72,75,81,85*,87,88,90*,94,95,100,102*
Comment: Pretty dense, given that 2 modulo 3 is largely excluded.
Base 86: x^5+67x^4+69x^3+31x^2+56x+29
1,2,3*,4,5*,6,9*,10,12,13,14,15*,16,17,18*,20,22,24,25*,26,27,28,29,30,31*,34,36,39,40,41*,43*,44,46,49,
52*,54*,57,58,61,62,63*,66,68,69,71,75,76,77*,80,81,82,83,84*,86*,88*,91,92,94,95*,99,101,103,105,107*
Comment: Incredibly dense.
Base 87: x^5+59x^4+74x^3+41x^2+25+1
1,2*,3,4,5,6,7,8,9,10,11*,12,14*,15,16,17,19,20,21,22,25,26,29*,32,34,35,39,40,41,42,47*,48,50*,53,
54*,55,56,59,61,62*,66,69,71,72*,74*,75*,76,77*,78,80,81*,86,87*,89*,90,92,93,95,98,99,101,102*
Comment: Again, very very dense, especially since 1 modulo 3 generates multiples of 3.
Base 88: x^5+52x^4+25x^3+74x^2+22x+1
2,4*,5,6,7,9*,10*,11,13,14,15,17,19,20*,23*,24,25,28*,30,32,33,39,40,45,48,
49,52,53,54,56,59*,63,66,67,70,72,74,75*,79,83,84,88*,90*,95*,96,97,99,100*
Comment: Fairly dense.
Base 89: x^5+45x^4+8x^3+31x^2+13x+79
1,3,4,5,6,8,9,11,14*,18,19,21,23,26*,29,30,33,36,39,44*,45,47,48,50,51,53,54*,56,
63,64,65,66,78,80,84*,86,89*,93*,96,99*,100,105,107,108,114,123,125,126,131*
Comment: Unremarkable.
Base 90: x^5+38x^4+20x^3+84x^2+71x+43
1*,2*,3,4,5*,6,8,9*,11,12,13*,15,16,17,18,19,20,22*,23*,26*,27*,28*,31,32*,34,35*,37*,38*,42,44,45*,46,48*,49,51,52,53,54,
55*,56,60*,61,62*,63,64,65*,66,67*,69,70,71,73*,74*,76*,78,79*,81,82*,83,84*,87*,88,89,90*,91,93*,94,95*,96,97,99,100,101*
Comment: Both enormously dense and with a lot of prime clustering.
Base 91: x^5+31x^4+62x^3+57x^2+89x+73
1*,3*,4,6,7,9*,10,13,15*,18,21,24,25*,28*,31,33,34,36,37,45,46,48*,49*,51,55,57,
58*,64,66,69,70*,72,78,79,82,84*,87,88*,90,91*,93,94,97,105,106,108,111,112*
Comment: Not very remarkable.
Base 92: x^5+25x^4+40x^3+47x^2+42x+9
2*,4,14,16,20,22*,28,32,40,50*,58,64*,68,76,86,88,92*,100*
Comment: Nothing special here.
Base 93: x^5+19x^4+43x^3+66x^2+65x+91
2,3*,5*,8*,9*,11,18,20,23*,24*,27,33*,38,44*,45,50,54,59,60*,68,74,78,80*,87,93*,94,99*,101,102,110,113,120*
Comment: Pretty meagre.
Base 94: x^5+13x^4+71x^3+40x^2+85x+67
1*,2,4,5*,6,9,10*,11,12,13,14,15,18,20,21*,24*,25*,29*,30,31,34*,35,36,39*,
40,44,47,49,50,51,55,61*,64,66,68,69,70,72,76,77,81,88,90,91*,94*,95,100*
Comment: Little cluster of primes.
Base 95: x^5+8x^4+27x^3+86x^2+75x+58
3*,5*,7,17*,23,29,33*,35,39,43,45,47,57,63*,65*,71,72,93*,95*,99*,105,107,113*,115,117*
Comment: Fairly dense among plausible values, string of three primes.
Base 96: x^5+3x^4+5x^3+42x^2+53x+49
2*,3,5,6,8*,12,15,17,21,23,26*,29,33,36*,38,45*,47,48,51,61,65*,66,68*,71,77,78*,80*,81,83,87,93*,95,96*,101*
Comment: Unremarkable.
This listing derives from taking the designation of a particular hootoo article, A41078748, and reading it as a base-13 number. If there really is something remarkable about it, it would have to be said that I was mediating or channelling something to come up with it. This brings me to my final note about the listing. In my investigation of the base-84 case, I went through the process initially with a multiplication sign in place of the exponentiation in the cube term, so that I was really investigating the polynomial 2x^5+71x^2+308x+73. The unbelievable thing is that while the correct polynomial generates primes and almost-primes at about as high a frequency as possible, the falsely substituted polynomial is even more extreme in this direction. Was I responding to some external stimulus in making my error?
Web-based factoring resource: http://www.alpertron.com.ar/ECM.HTM
