Can we find integers x and y such that f,g,h are strictely positive integers Let a>2 and b>2 two strictely positive integers. Let us consider the following quantities: f= (xy+ay+a^(2))/(by) g=a(y+a)(xy+ay+a^(2))/(by^(2)x) h=(y+a)/(b)

Raiden Barr 2022-10-25 Answered
Can we find integers x and y such that f,g,h are strictely positive integers
Let a > 2 and b > 2 two strictely positive integers. Let us consider the following quantities:
f = x y + a y + a 2 b y
g = a ( y + a ) x y + a y + a 2 b y 2 x
h = y + a b
My question is:
Can we find integers x and y (not necessarly positive) such that f,g,h are strictely positive integers. Or at lest how one can proves that they are exist.
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Answers (1)

rcampas4i
Answered 2022-10-26 Author has 22 answers
If we let,
a = ( b k 1 ) y , x = b y
then the three polynomials lose their denominators,
( b k 2 k + 1 ) y , ( b k 1 ) ( b k 2 k + 1 ) k y , k y
and are positive integers if b , k , y are positive integers.
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