Marco Damele

Tag: teoria dei numeri

  • The Diophantine equation X⁴-Y⁴-X=0

    —

    by

    Marco Damele
    in

    Let’s solve the Diophantine equation X⁴-Y⁴-X=0 together so as to see a standard approach that is encountered in solving Diophantine equations of the form Z(f(Z)-1)=Y² since f is a function of Z. For any typos I invite you to report them in the comments. Let’s suppose that (X,Y) is a non-trivial solution, that is, X…

  • A simple Diophantine equation

    —

    by

    Marco Damele
    in

    Let’s solve a Diophantine equation together using only elementary considerations. Suppose we are asked to determine all the integers X and Y such that X³-X+1=3Y⁵. As is usually done, we assume that such a solution exists, let’s say (X,Y), and we try to understand if we can make the value of X and Y explicit…

  • A Diophantine equation

    —

    by

    Marco Damele
    in

    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…

  • Equations in algebraically closed fields.

    —

    by

    Marco Damele
    in

    Let K be an algebraically closed field and f∈K[X,Y] non-constant. Let us prove that f has infinitely many zeros in K x K. Let us define the following sets: A={a ∈K : f(a,y) is constant in K[y]}B={b∈K : f(x,b) is constant in K[x] } if one of them is finite we are done. In fact…

  • Sum of squares in finite fields

    —

    by

    Marco Damele
    in

    Let K be a finite field. Let us prove that every element of K is the sum of two squares. Let us distinguish two cases. K has characteristic 2. Let us consider the map Φ:K—>K, x |—>x² . Let us observe that Φ is an injective homomorphism. In fact it is a homomorphism having the…

  • Fermat Last theorem on ℚ(√2)

    —

    by

    Marco Damele
    in

    Does Fermat’s Last Theorem hold in ℚ(√2)? No, in fact it can be shown that if n>3 then the equation Xⁿ+Yⁿ=Zⁿ has no non-trivial solutions in ℚ(√2,√3) (therefore not even in ℚ(√2) however for n=3 we have that: (18+17√2)³+(18-17√2)³=42³ Source: https://eprints.whiterose.ac.uk/194494/14/Fermat.pdf, https://math.stackexchange.com/questions/4438180/does-fermats-last-theorem-imply-sqrt2-not-in-mathbbq