p will divide the difference f(x) - g(x) = 2x^3 + 10x^2 + 2.
Both functions are always odd for integers so they're divisible only by odd primes and the factor 2 is irrelevant. Divide it out and subtract the result from f(x) to get x^3(x+4)(x+1).
x<p so the factor x^3 is irrelevant too. Then the possibilities are x=-4 or x=-1 mod p.
Apply those values to find f(x)=5 or 17 mod p and g(x)=-5 or -17 mod p. As the residues are prime, only p=5 or p=17 work.
Checking, f(4)=2705=5*541, g(4)=2415=5*483, f(13)=525929=17*30937 and g(13)=519843=17*30579.
The primes are thus 5 and 17 with sum 22.
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Posted by xdog
on 2018-11-17 21:05:43 |
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