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

= 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
 * $$111862\cdot3^{36}-1=16789886093592915673301.$$
 * --Lambiam 08:53, 19 June 2023 (UTC)
 * WolframAlpha confirms that both values are indeed prime. GalacticShoe (talk) 16:00, 19 June 2023 (UTC)