I am a visitor of this beautiful site, my name is Angelo Spina, I would like to resolve the three following problems, in fact after many attempts I have not succeeded in it, for this reason I kindly ask you to give me a help.(adsbygoogle = window.adsbygoogle || []).push({});

PROBLEM 1.

If the equation y² + a p² = 2 x² (where a is a positive integer, p is an odd prime number) admits a solution (y,x) of integers with gcd(y,x)=1, how can I prove that the equation y² + p² = 2 x² also admits a solution (y,x) of integers with gcd(y,x)=1 ?

PROBLEM 2.

If the two equations y² + p² = 2 x² , y² + q² = 2 x² (where p and q are odd prime numbers) respectively admit the integer solutions (y',x') and (y '', x '') with gcd(y',x')=1 and gcd(y'',x'')=1, how can I prove that the equation y² + (pq)² = 2 x² also admits at least an integer solution (y*,x *) with gcd(y*,x*)=1 ? Is it possible to find a formula that allows to obtain (y*,x *) from the knowledge of (y',x') and (y '', x '') ?

PROBLEM 3.

If the equation y² + n² = 2 x² (where n is a positive integer greater than 3) admits integer solutions con gcd = 1, how can I prove that the equations

y² + p² = 2 x², y² + q² = 2 x², y² + r² = 2 x², y² + s² = 2 x²,......, (where p, q, r, s,....are the prime factors of n) also admit integer solutions con gcd=1?

Certain of your courtesy, I thank you very much.

Angelo.

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Equation in Z

Loading...

Similar Threads - Equation | Date |
---|---|

I Solutions to equations involving linear transformations | Mar 6, 2018 |

I Solving System of Equations w/ Gauss-Jordan Elimination | Sep 18, 2017 |

I Solving a system of linear equations using back substitution | Aug 30, 2017 |

**Physics Forums - The Fusion of Science and Community**