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angelo
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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.
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.
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.