Mersenne numbers: if equation x + y + axy = b has no soln's in positive integers, a =

Diego_D

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True or False:
If the equation x + y + axy = b has no solution in positive integers, where ab + 2 = 2a, and a - a prime, then 2a -1 - prime.

Decision:
x + y + axy = b
ax + ay + ax . ay + 1 = ab + 1
(ax + 1) (ay + 1) = ab + 1
(ax + 1) (ay + 1) = ab + 2-1
(ax + 1) (ay + 1) = 2a -1
All would be good if done with this.
Can I find an answer in the discussions? I have some doubts that the equation has no solution in positive integers, but 2a -1 - composite number.
 
Suppose a = 2. Then (2x+1)(2y+1)=221=3\displaystyle (2x+1)(2y+1)=2^2-1=3 and it is impossible for both x and y to be positive since one of the factors on the LHS must be 1. Also 221=3\displaystyle 2^2-1=3 is prime. So the statement is true for a = 2.

Now let a be an odd prime. I will use this result: if p is a prime dividing 2a1\displaystyle 2^a-1, then p1(moda)\displaystyle p\equiv1\pmod a. It is not hard to prove it but I shall leave it for now.

Clearly the product of two primes dividing 2a1\displaystyle 2^a-1 is also 1(moda)\displaystyle \equiv1\pmod a. It follows that if p is not the only prime dividing 2a1\displaystyle 2^a-1 or divides it more than once, we could choose x and y such that ax+1=p\displaystyle ax+1=p and ay+1\displaystyle ay+1 is the product of the other prime factors, and then x and y would be positive solutions to the given equation. Hence p must be the only prime factor of 2a1\displaystyle 2^a-1 dividing it just once, meaning 2a1=p\displaystyle 2^a-1=p is prime.
 
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