A059770 -id:A059770 - cp's OEIS frontend

A242488 Triangle read by rows in which row n lists numbers k such that the greatest prime factor of k^2 - 2 is A038873(n), the n-th prime not congruent to 3 or 5 mod 8.

2, 3, 4, 10, 6, 11, 45, 108, 5, 18, 28, 74, 156, 235, 8, 23, 39, 116, 1201, 17, 24, 58, 147, 304, 550, 2272, 390050, 7, 40, 54, 87, 101, 181, 557, 1558, 43764, 314766, 12, 59, 130, 225, 414, 1077, 1124, 2686, 3420, 4035, 32, 41, 178, 333, 698, 844, 1638, 4567, 15362, 364384

Offset: 1

Juri-Stepan Gerasimov, May 16 2014

From Andrew Howroyd, Dec 22 2024: (Start)
For any prime p, there are finitely many x such that x^2 - 2 has p as its largest prime factor.
The Filip Najman data file gives all 537 numbers x such that x^2 - 2 has no prime factor greater than 199. This includes a value for x = 1 which is not included here. (End)

			Triangle of numbers k such that p is the greatest prime factor of k^2 - 2:
p\k  |  1 |  2 |  3  |  4  |  5   |  6   |  7   |  >= 8
------------------------------------------------------------------------
   2 |  2 |    |     |     |      |      |      |
   7 |  3 |  4 |  10 |     |      |      |      |
  17 |  6 | 11 |  45 | 108 |      |      |      |
  23 |  5 | 18 |  28 |  74 |  156 |  235 |      |
  31 |  8 | 23 |  39 | 116 | 1201 |      |      |
  41 | 17 | 24 |  58 | 147 |  304 |  550 | 2272 | 390050;
  47 |  7 | 40 |  54 |  87 |  101 |  181 |  557 | 1558, 43764, 314766;
  71 | 12 | 59 | 130 | 225 |  414 | 1077 | 1124 | 2686, 3420, 4035;
  73 | 32 | 41 | 178 | 333 |  698 |  844 | 1638 | 4567, 15362, 364384;
  ...
6 is a term of row 3 because (6^2 - 2)/17 = 2 and 2 < 17;
11 is a term of row 3 because (11^2 - 2)/17 = 7 and 7 < 17;
45 is a term of row 3 because (45^2 - 2)/17^2 = 7 and 7 < 17;
108 is a term of row 3 because (108^2 - 2)/17 = 686 = 2*7^3 and 7 < 17.

Andrew Howroyd, Table of n, a(n) for n = 1..536 (first 21 rows for primes up to 199)
Filip Najman, Smooth values of some quadratic polynomials, Glasnik Matematicki Series III 45 (2010), pp. 347-355.
Filip Najman, List of Publications Page (Adjacent to entry number 7 are links with a data file for the first 21 rows of this sequence).

Cf. A038873, A164314, A059770 (first terms for n>1), A185396 (last terms), A379348 (row lengths).

Cf. A223701.

Converted to triangle by Andrew Howroyd, Dec 22 2024

A059772 Smallest prime p such that n is a solution mod p of x^2 = 2, or 0 if no such prime exists.

Original entry on oeis.org

0, 7, 7, 23, 17, 47, 31, 79, 0, 17, 71, 167, 97, 223, 127, 41, 23, 359, 199, 439, 241, 31, 41, 89, 337, 727, 0, 839, 449, 137, 73, 1087, 577, 1223, 647, 1367, 103, 0, 47, 73, 881, 1847, 967, 0, 151, 2207, 1151, 2399, 1249, 113, 193, 401, 0, 3023, 1567, 191, 0, 71

Offset: 2

Klaus Brockhaus, Feb 21 2001

nonn

Solutions mod p are represented by integers from 0 to p-1. The following equivalences hold for n > 1: There is a prime p such that n is a solution mod p of x^2 = 2 iff n^2-2 has a prime factor > n; n is a solution mod p of x^2 = 2 iff p is a prime factor of n^2-2 and p > n. n^2-2 has at most one prime factor > n, consequently such a factor is the only prime p such that n is a solution mod p of x^2 = 2. For n such that n^2-2 has no prime factor > n (the zeros in the sequence), cf. A060515.

			a(11) = 17, since 11 is a solution mod 17 of x^2 = 2 and 11 is not a solution mod p of x^2 = 2 for primes p < 17. Although 11^2 = 2 mod 7, prime 7 is excluded because 7 < 11 and 11 = 4 mod 7.

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A038873, A059770, A059771, A060515.

f:= proc(n) local P;
  P:= select(`>`,numtheory:-factorset(n^2-2),n);
  if P = {} then 0 else min(P) fi
end proc:
map(f, [$2..100]); # Robert Israel, Feb 23 2016

a[n_] := Module[{P}, P = Select[FactorInteger[n^2 - 2][[All, 1]], # > n&]; If[P == {}, 0, Min[P]]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Apr 10 2019, from Maple *)

If n^2-2 has a (unique) prime factor p > n, then a(n) = p, else a(n) = 0.

Offset corrected by R. J. Mathar, Aug 21 2009

A059771 Second solution of x^2 = 2 mod p for primes p such that a solution exists.

Original entry on oeis.org

0, 4, 11, 18, 23, 24, 40, 59, 41, 70, 64, 83, 65, 62, 111, 106, 105, 154, 134, 141, 179, 208, 148, 140, 219, 197, 153, 175, 149, 245, 193, 311, 186, 340, 288, 246, 348, 312, 243, 227, 418, 419, 377, 260, 292, 396, 346, 272, 368, 543, 451, 433, 379, 413, 321

Offset: 1

Klaus Brockhaus, Feb 21 2001

Solutions mod p are represented by integers from 0 to p-1. For p > 2: If x^2 = 2 has a solution mod p, then it has exactly two solutions and their sum is p; i is a solution mod p of x^2 = 2 iff p-i is a solution mod p of x^2 = 2. No integer occurs more than once in this sequence. Moreover, no integer (except 0) occurs both in this sequence and in sequence A059770 of the first solutions (Cf. A059772).

			a(6) = 24 since 41 is the sixth term of A038873, 17 and 24 are the solutions mod 41 of x^2 = 2 and 24 is the larger one.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A038873, A059770, A059772.

R:= 0: p:= 2: count:= 1:
while count < 100 do
  p:= nextprime(p);
  if NumberTheory:-QuadraticResidue(2,p)=1 then
    v:= NumberTheory:-ModularSquareRoot(2,p);
    R:= R, max(v,p-v);
    count:= count+1
  fi
od:
R; # Robert Israel, Sep 07 2023

a(n) = second (larger) solution of x^2 = 2 mod p, where p is the n-th prime such that x^2 = 2 mod p has a solution, i.e. p is the n-th term of A038873. a(n) = 0 if x^2 = 2 mod p has one solution (only for p = 2).

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A242488 Triangle read by rows in which row n lists numbers k such that the greatest prime factor of k^2 - 2 is A038873(n), the n-th prime not congruent to 3 or 5 mod 8.

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A059772 Smallest prime p such that n is a solution mod p of x^2 = 2, or 0 if no such prime exists.

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A059771 Second solution of x^2 = 2 mod p for primes p such that a solution exists.

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