Positive Integer values of a fraction I am trying to find all strictely positive integers x,y so that z=(4x+1)/(4y-1) is a strictely positive integer.

Deon Moran

Deon Moran

Answered question

2022-10-28

Positive Integer values of a fraction
I am trying to find all strictely positive integers x , y so that
z = 4 x + 1 4 y 1
is a strictely positive integer.
Some cases are possible when choosing special values of x and y. However, I am not able to find the general solution.

Answer & Explanation

Momellaxi

Momellaxi

Beginner2022-10-29Added 14 answers

(This is not the full answer), but we have at least infinite pairs ( x , y ) N 2 for which z is positive integer. Say y is arbitrary, then x = 4 t y t y will solve the equation where t is arbitrary (positive) integer.
Actually, I just realized that are all solutions of the given equation.
Rewrite like this
4 z y z = 4 x + 1 4 z + 1 z = 4 t 1
So we get x = 4 t y y t ( ). So, however I chose t and y and x is of form ( ), then z will be (positive) integer.

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