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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)[reply]

See Catalan's conjecture. --RDBury (talk) 11:42, 22 July 2016 (UTC)[reply]

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)[reply]

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)[reply]

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

July 22

Does every natural n {\displaystyle n} satisfy that n 2 + 1 {\displaystyle n^{2}+1} cannot be any power of any natural number - with a natural exponential bigger than 1?

Does every natural $n$ satisfy that $n^{2}+1$ cannot be any power of any natural number - with a natural exponential bigger than 1?