Two Math Conjectures

Hehe… I’m not quite sure if these are already conjectures, or if they are false, but I made two conjectures anyways.

1. Base conversion (pretty sure about this one):
   • oldbase number in newbase = log(newbase) number * 10
   • Ex: 2 = log(2) 2 * 10 = 10
2. Prime Numbers (not so sure):
   • x^2 - x + y is prime, if x and y are prime and x ≠ y
   • Ex: 2^2 - 2 + 3 = 5
3. Highly Composite Numbers and the nearest prime number have a difference that is always prime

Please find counter-examples or even prove them… EDIT: Freakman found a counter example for #2, 344^2 - 344 + 653 = 118645. So now both x and y have to be prime. That should work, né?

EDIT: Seems like the third one was proven (it wasn’t mine, it was Bill R. McEachen’s (billymac00 AT excite.com)). Nice!

More edits! This post, so far, has gotten the most publicity, so I have moved it up into the near-past so it remains on the first page. Originally posted 22:30 11/26/2005. Also, I presume Conj #1 is part of the rule for base conversion… Logs are used for these kinds of things.