Marco Damele

Tag: Diofantea

  • A Diophantine sistem.

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    Marco Damele
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    We want to determine all integers x₁,…,x₂₀₁₁ such that If such integers exist, by Fermat’s little theorem we would have that xi⁷=xi (mod 7) for every i=1,…,2011. So 123=321 (mod 7) or 4=6 (mod 7) which is absurd.

  • The Diophantine equation 4X⁴-16X³+20X²-8X-3=Y²

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    Marco Damele
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    Let’s solve the Diophantine equation: 4X⁴-16X³+20X²-8X-3=Y². Let’s suppose that X,Y are two integers satisfying the equation then: (2X²-4X-1)(2X²-4X+3)=Y². Let’s observe that 2X²-4X-1 and 2X²-4X+3 are coprime since if p is a possible common prime divisor it must be odd, let’s say it is p and we would have that -4=0 mod p therefore p=2, absurd.…

  • A Diophantine equation

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    Marco Damele
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    Let us determine all the prime positive p and q such that p²-2q²=1. We observe that p=3 and q=2 satisfy the question since they are both positive primes and (3)²-2(2)²=9-8=1. Conversely, if p and q are prime positives such that p²-2q²=1 then p²=2q²+1 therefore p² is odd or p is odd let’s say p=2k+1 then…