Wikipedia:Reference desk/Archives/Mathematics/2016 July 22

= July 22 =

Does every natural $$n$$ satisfy that $$n^2+1$$ cannot be any power of any natural number - with a natural exponential bigger than 1?
Please notice, that if the " + " is replaced by " - ", then the conjecture is incorrect: Check: n=3. Additionally, if the " 2 " is replaced by another natural number, then the conjecture is unnecessarily correct. Check: replacing " 2 " by " 1 " while n=3, or replacing " 2 " by " 3 " while n=2. HOTmag (talk) 11:22, 22 July 2016 (UTC)


 * See Catalan's conjecture. --RDBury (talk) 11:42, 22 July 2016 (UTC)
 * You kept changing the question while I wrote answers and then I got beaten. Catalan's conjecture, proved by Preda Mihăilescu, says 23 + 1 = 32 is the only case of powers one apart. PrimeHunter (talk) 11:45, 22 July 2016 (UTC)
 * Didn't mean to step on your answer, sorry. Mihăilescu's proof is pretty high powered so I wonder if there is an elementary proof for this special case, or for the case n2+1 = m3. --RDBury (talk) 11:54, 22 July 2016 (UTC)

Thank you RDBury, and thank you PrimeHunter. HOTmag (talk) 11:48, 22 July 2016 (UTC)