If x and y are integers >= 2, and ${\displaystyle gcd(x^{2}-1,y^{2}-1)=1}$, and for every r > 1, x and y are not both perfect r-th powers, then is there always a prime of one of these four forms: ${\displaystyle x\cdot y^{n}+1}$, ${\displaystyle x\cdot y^{n}-1}$, ${\displaystyle y\cdot x^{n}+1}$, ${\displaystyle y\cdot x^{n}-1}$, with n >= 1? 36.233.231.45 (talk) 04:31, 19 June 2023 (UTC)[reply]

Such things, if true, tend to be hard to prove. For the case ${\displaystyle x=3,y=13022,}$I found that the least value of ${\displaystyle n}$to give a prime is ${\displaystyle 24}$:

${\displaystyle 13022\cdot 3^{24}+1=3677797424055583.}$

(At least, that value is a probable prime.)  --Lambiam 08:32, 19 June 2023 (UTC)[reply]

This can be increased to

${\displaystyle 111862\cdot 3^{36}-1=16789886093592915673301.}$

 --Lambiam 08:53, 19 June 2023 (UTC)[reply]

WolframAlpha confirms that both values are indeed prime. GalacticShoe (talk) 16:00, 19 June 2023 (UTC)[reply]