Find the largest prime factor of 257^4 + 32^3 - 8193^2 - 640^2.
I believe this is rouughly how it is supposed to be solved. The given expression:
=(2^8+1)^4+(2^5)^3-(2^13+1)^2-(2^9+2^7)^2
=2^8(2^24-[2^18+4*2^16][-2^10+4*2^8]+[2^7-2^6-2*2^5]+4)
= 2^8(2^24+4) after cancelling the matching terms in square brackets.
We can ignore 2^8 since its only prime factor is 2, leaving (2^24+4), or 4(2^22+1)
We can ignore the 4, since again its only prime factor is 2, leaving (2^22+1) or 4194305.
That number is clearly divisible by 5, but before we do that, note that if 2^n+1 is divisible by some prime, p, then p is divisible by 2n with remainder 1, i.e. here of the form 44n+1, for p>44.
Such numbers include {45,89,133,177,221,265,309,353,397...} but many of these can be eliminated as not prime, leaving {89,353,397,..} A142292 in Sloane.
(2^22+1)/397 = 10565. 10565/5 = 2113, while 2113 mod44=1, and indeed 2113 is also prime, and the largest prime factor of the expression first given.
So the 4 primes are 2,5,397, and 2113.
Checking: 4294968320=2^10×5×397×2113
Edited on December 13, 2024, 5:38 am
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Posted by broll
on 2024-12-13 05:37:50 |
Possible Solution
