A257231 - OEIS

%I #25 Sep 08 2022 08:46:12

%S 1,1,4,1,4,1,5,9,4,1,4,1,16,9,4,1,4,1,16,9,4,1,7,25,16,9,4,1,4,1,36,

%T 25,16,9,4,1,16,9,4,1,4,1,16,9,4,1,36,25,16,9,4,1,36,25,16,9,4,1,4,1,

%U 36,25,16,9,4,1,16,9,4,1,4,1,36,25,16,9,4,1,16,9,4,1,36,25,16,9,4

%N a(n) = n^2 mod p where p is the least prime greater than n.

%C Conjecture: a(n) is always a positive square, except for the terms 5, 7, 69, 42 and 17 given by n = 7, 23, 113, 114, and 115 respectively. It is easy to show that nonsquare terms are in [p, q) iff p and q are consecutive primes and q-p > sqrt(q). There are no gaps between consecutive primes greater than sqrt(q) for 127 < q < 4*10^18 (see Nicely's table of maximal prime gaps).

%H Chris Boyd, <a href="/A257231/b257231.txt">Table of n, a(n) for n = 1..10000</a>

%H Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/gaps/gaplist.html">First occurrence prime gaps</a> [For local copy see A000101]

%H Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/gaps.html">Gaps between consecutive primes</a>

%e a(23) = 7 because 23^2 mod 29 = 7.

%e a(24) = 25 because 24^2 mod 29 = 25.

%t Table[Mod[n^2, NextPrime@ n], {n, 87}] (* _Michael De Vlieger_, Apr 19 2015 *)

%t Table[PowerMod[n,2,NextPrime[n]],{n,90}] (* _Harvey P. Dale_, May 24 2015 *)

%o (PARI) a(n)=n^2%nextprime(n+1)

%o (Magma) [n^2 mod NextPrime(n): n in [1..80]]; // _Vincenzo Librandi_, Apr 19 2015

%Y Cf. A257230.

%K nonn,easy

%O 1,3

%A _Chris Boyd_, Apr 19 2015