h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

How many non-primes are equal to the sum of their prime divisors?

Answer, only one that I know of, 2 x 2 = 4, 2 + 2 = 4

There may or may not be another number like that in all infinity. I simply don't know.

I'm pretty sure there isn't any other such number, as in general multiplication makes things bigger than addition does. For example, 643 + 1023 is equal to 1666, but when multiplied together, they make something much bigger: 657,789.

I think that is a possibility that there may be other numbers of the type, especially up in the range between Skewes' numbers 1 & 2 or higher, but cannot rule out the possibility even with smaller, but still very large numbers.

Something of an academic point since these are numbers of a size such that one could not establish a one to one correspondence against material objects even if all the mensurable particles of the entire Universe were put to the purpose. ...one runs out of things to count before one runs out of counting numbers.

http://www.daviddarling.info/encyclopedia/S/Skewes_Number.html

I'd guess that I could prove that there are no such numbers in two or three lines of proof.

Yes, it's easy to prove.

Call the numbers a and b. If one of the numbers, say a, is equal to 1, then we have:

1 x b = 1 + b
=>
b = 1 + b
=>
0 = 1

which is impossible, so neither of the numbers can be 1. They must both be greater than 1.

If one of the numbers, say a, is equal to 2, then we have:

2 x b = 2 + b
=>
b = 2

so both the numbers are equal to 2. This is one possible solution:

2 x 2 = 2 + 2

If both the numbers are greater than 2, then:

a x b = a + b
=>
a x b - b = a
=>
b(a - 1) = a

Since a > 2, a-1 > 1. So a is b times a number greater than 1, so a is greater than b.

Similarly,

a x b = a + b
=>
a x b - a = b
a(b - 1) = b

so b is greater than a.

So a>b and b>a. Since this is clearly impossible, there are no other solutions besides the one we've already found.

Oh, that only proves it for when the number has only two divisors. For the general case it would be a bit more complicated, but I'm pretty sure that it could be proved.

I'm still on my summertime nocturnal schedule (against the 43 - 48 degree summer temperatures of the Coachella Valley), it's 6:11am PST now and I've been up since about 4:00pm yesterday.

Your proof looks good, but I'll want to check it again this afternoon.

I believe that 4 is the only number equal to the sum and the product of its prime divisors. Certainly the probability of such a number drops off precipitously and astronomically with increasing size.

A generalized proof has always eluded me whenever I've attempted it
though.

Meanwhile I've always found contradictions of the kind represented in the conclusion of your proof amusing in their own right.