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Powered factors (Posted on 2024-12-12)

Given that there are exactly four primes that divide the number

257⁴ + 32³ - 8193² - 640²

find the largest of the four primes.

No Solution Yet Submitted by Danish Ahmed Khan

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re: Possible Solution

Comment 3 of 3 |

(In reply to Possible Solution by broll)

Nice work!

I do have one addendum, on how to factor 2^24+4.

Here I will not factor out the 4 immediately, the +4 at the end is actually quite useful for a lesser known factorization: x^4+4 = (x^2+2x+2)*(x^2-2x+2)

Let x=2^6. Then 2^24+4 = (2^12+2^7+2)*(2^12-2^7+2) = (4096+128+2)*(4096-128+2) = 4226*3970.

4226 = 2*2113 and 3970 = 2*5*397, so the four prime factors of 2^24+4 are 2, 5, 397, and 2113.

Posted by Brian Smith on 2024-12-13 12:39:27

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