Testwiki:Reference desk/Archives/Mathematics/2023 June 19

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June 19

Is there always a prime of this form?

If x and y are integers >= 2, and $gcd(x^2-1,y^2-1)=1$, 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: $x\cdot y^n+1$, $x\cdot y^n-1$, $y\cdot x^n+1$, $y\cdot x^n-1$, with n >= 1? 36.233.231.45 (talk) 04:31, 19 June 2023 (UTC)

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

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

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

This can be increased to

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

 --Lambiam 08:53, 19 June 2023 (UTC)

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