A063988 Triangle in which n-th row gives quadratic non-residues modulo the n-th prime.

2, 2, 3, 3, 5, 6, 2, 6, 7, 8, 10, 2, 5, 6, 7, 8, 11, 3, 5, 6, 7, 10, 11, 12, 14, 2, 3, 8, 10, 12, 13, 14, 15, 18, 5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 22, 2, 3, 8, 10, 11, 12, 14, 15, 17, 18, 19, 21, 26, 27, 3, 6, 11, 12, 13, 15, 17, 21, 22, 23, 24, 26, 27, 29, 30, 2, 5, 6, 8, 13, 14

Offset: 2

Author

Suggested by Gary W. Adamson, Sep 18 2001

Keywords

• nonn
• tabf
• nice
• easy

Examples

			Mod the 5th prime, 11, the quadratic residues are 1,3,4,5,9 and the non-residues are 2,6,7,8,10.
Triangle begins:
  2;
  2, 3;
  3, 5, 6;
  2, 6, 7, 8, 10;
  ...
		

References

• Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See Table 82 at p. 202.

Links

• T. D. Noe, Rows n=2..100 of triangle, flattened

Crossrefs

Cf. A063987.

Programs

Maple

with(numtheory): for n from 1 to 20 do for j from 1 to ithprime(n)-1 do if legendre(j, ithprime(n)) = -1 then printf(`%d,`,j) fi; od: od:

Mathematica

row[n_] := Select[p = Prime[n]; Range[p - 1], JacobiSymbol[#, p] == -1 &]; Table[row[n], {n, 2, 12}] // Flatten (* Jean-François Alcover, Oct 17 2012 *)

PARI

residue(n,m)={local(r);r=0;for(i=0,floor(m/2),if(i^2%m==n,r=1));r}
  isA063988(n,m)=!residue(n,prime(m)) \\ Michael B. Porter, May 07 2010

PARI

tabf(nn) = {for(n=1, prime(nn), p = prime(n); for (i=2, p-1, if (kronecker(i, p) == -1, print1(i, ", "));); print(););} \\ Michel Marcus, Jul 19 2013

Python

from sympy import jacobi_symbol as J, prime
def a(n):
    p=prime(n)
    return [i for i in range(1, p) if J(i, p)==-1]
print([a(n) for n in range(2, 13)]) # Indranil Ghosh, May 27 2017

Extensions

More terms from James Sellers, Sep 25 2001