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A254328
Numbers k such that all x^2 mod k are squares (including 0 and 1).
4
1, 2, 3, 4, 5, 8, 12, 16
OFFSET
1,2
COMMENTS
Are there any more terms > 16?
There are no more terms less than 10^12. Probably the sequence is finite. - Charles R Greathouse IV, Jan 29 2015
This is a subsequence of A303704, so it is full. - Jianing Song, Feb 14 2019
EXAMPLE
Terms k <= 16 and the squares mod k:
1: [0]
2: [0, 1]
3: [0, 1, 1]
4: [0, 1, 0, 1]
5: [0, 1, 4, 4, 1]
8: [0, 1, 4, 1, 0, 1, 4, 1]
12: [0, 1, 4, 9, 4, 1, 0, 1, 4, 9, 4, 1]
16: [0, 1, 4, 9, 0, 9, 4, 1, 0, 1, 4, 9, 0, 9, 4, 1]
k = 10 is not a term: in the list of squares mod 10, [0, 1, 4, 9, 6, 5, 6, 9, 4, 1], the numbers 5 and 6 are not squares.
MATHEMATICA
f[n_] := Mod[Range[n]^2, n]; Select[Range@ 10000, AllTrue[f@ #, IntegerQ[Sqrt[#]] &] &] (* AllTrue function introduced in version 10; Michael De Vlieger, Jan 29 2015 *)
PROG
(PARI) isok(n)=for(k=2, n-1, if(!issquare(lift(Mod(k, n)^2)), return(0))); return(1);
for(n=1, 10^9, if(isok(n), print1(n, ", ")));
(PARI) is(n)=for(k=sqrtint(n)+1, n\2, if(!issquare(k^2%n), return(0))); 1
for(m=10, 10^6, for(k=0, sqrtint(2*m), if(is(t=m^2-k^2), print(t))))
\\ Charles R Greathouse IV, Jan 29 2015
CROSSREFS
KEYWORD
nonn,fini,full
AUTHOR
Joerg Arndt, Jan 28 2015
EXTENSIONS
Keywords fini and full added by Jianing Song, Feb 14 2019
STATUS
approved