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

"the odd perfect number, if it exists, will be expressible in the form i^a.j^b.k^c... where i, j, k... and a are all in the form 4n+1 (that is, one more than a multiple of 4) and b, c... are all even."

Actually, the condition is that i and a are of the form 4n+1 and b, c,... are even. j, k,... can be of the form 4n+1 or 4n+3.

All odd perfect numbers must have their largest prime factor greater than 10,000,000 (Jenkins) and third largest factor at least 100 (Ianucci).

No, you're wrong there. Euler proved that the condition is as I have stated it.

Your statement, that j, k etc can be of the form 4n+1 or 4n+3 just means that they are odd numbers. Euler's statement imposes a further restriction on them.

I can't verify your statement about the largest prime factor - I've seen it stated that it must be greater than 300,000. You statement about the second largest prime factor is certainly true - in fact it must be greater than 1000.