A358404 Multipliers involving Fibonacci-like sequences and Pythagorean triples.

2, 3, 5, 8, 13, 21, 23, 34, 41, 61, 85, 89, 144, 233, 255, 264, 377, 383, 397, 443, 610, 762, 875, 987

Offset

1,1

Comments

A positive integer m is an element of this sequence if and only if there exists a Pythagorean triple of the form (m*G_0, m*G_1, G_n), where (G_k) is a Fibonacci-like sequence, i.e., a sequence with arbitrary positive integer starting values G_0 and G_1 and satisfying the recurrence G_k = G_{k-1} + G_{k-2} for every index k > 1.

Links

Table of n, a(n) for n=1..24.

Formula

Each element of this list has a unique representation of the form m = m(c, j) = G_n / c, where j is an arbitrary nonnegative integer and c is "good", meaning that all of its prime divisors are of the form 4k + 1 and the Fibonacci entry point t of c is odd, in which case n = ((2j + 1)t + 1)/2 and (G_0, G_1, c) is the unique primitive Pythagorean triple such that G_0/G_1 is congruent to F_n/F_{n-1} modulo c.

Example

m = m(5, 0) = 2, since the Fibonacci-like sequence (G_n) with G_0 = 4 and G_1 = 3 has G_3 = 10 and (m*G_0, m*G_1, G_3) = (8, 6, 10) is a Pythagorean triple. Since m = 2 is the smallest positive integer with this property, m(1) = 2.

Mathematica

(* Fibonacci entry point *)
T[m_] :=
  Module[{fi = FactorInteger[m], lenN, i, fi2, p, e, q, n1, divs,
    nDivs, d, found, preres, result = 1},
   If[m == 1, Return[1]];
   len = Length[fi];
   {p, e} = fi[[1]];
   q = p^e;
   If[len == 1,
    If[p == 5, Return[q]];
    If[e == 1,
     result = p - JacobiSymbol[p, 5];
     While[EvenQ[result] && Mod[Fibonacci[result], m] == 0,
      result /= 2];
     If[Mod[Fibonacci[result], m] != 0, result *= 2];
     fi2 = FactorInteger[result];
     If[EvenQ[result], Drop[fi2, 1]];
     n1 = Product[fi2[[i, 1]]^fi2[[i, 2]], {i, Length[fi2]}];
     divs = Divisors[n1];
     nDivs = Length[divs];
     found = False;
     For[i = 2, i <= nDivs && ! found, i++,
      d = divs[[i]];
      If[Mod[Fibonacci[d], m] == 0,
       found = True;
       result = d;
       Return[result];
       ];
      ],
     result = p^(e - 1 - If[p == 2 && e > 2, 1, 0])*T[p];
     Return[result];
     ],
    result = LCM[T[q], T[m/q]];
    ];
   result
   ];
(* Good moduli *)
GoodQ[m_] :=
  Module[{fi, len, i, p, t},
   If[m < 5, Return[False]];
   fi = FactorInteger[m];
   len = Length[fi];
   For[i = 1, i <= len, i++,
    p = fi[[i, 1]];
    If[Mod[p, 4] != 1, Return[False]];
    ];
   True
   ];
{* Great moduli *)
GreatQ[m_] := GoodQ[m] && OddQ[T[m]];
(* Fibonacci modular ratio *)
R[c_, k_] :=
  Module[{f0 = Fibonacci[k], f1},
   If[GCD[f0, c] > 1, Return[$Failed]];
   f1 = Fibonacci[k + 1];
   Mod[f1*PowerMod[f0, -1, c], c]
   ];
(* Starting pair for Fibonacci-like sequence *)
StartingPair[c_] :=
  Module[{pr, len, i, r0, t, n, r, u, v, g0, g1, preres},
   If[! GreatQ[c], Return[$Failed]];
   t = T[c];
   n = (t + 1)/2;
   r0 = R[c, n - 1];
   pr = PowersRepresentations[c, 2, 2];
   len = Length[pr];
   For[i = 1, i <= len, i++,
    {u, v} = pr[[i]];
    If[GCD[u, v] == 1,
     r = Mod[v*PowerMod[u, -1, c], c];
     preres = {Abs[u^2 - v^2], 2 u*v};
     If[r == c - r0, Return[preres]];
     If[r == r0, Return[Reverse[preres]]];
     ];
    ];
   $Failed
   ];
(* Great modulus multiplier *)
M[c_, j_] :=
  Module[{t, n0, n, g0, g1, result},
   If[! GreatQ[c], Return[$Failed]];
   {g0, g1} = StartingPair[c];
   t = T[c];
   n0 = (t + 1)/2;
   n = n0 + j*t;
   (g0*Fibonacci[n - 1] + g1*Fibonacci[n])/c
   ];
(* Master table *)
MasterTable[mMax_] :=
  Module[{c, j, m, g0, g1, t, n0, n, done, result = {}},
   For[c = 5, c <= GoldenRatio*mMax^2, c += 4,
    While[! GreatQ[c], c += 4];
    If[c <= GoldenRatio*mMax^2,
     {g0, g1} = StartingPair[c];
     t = T[c];
     n0 = (t + 1)/2;
     For[j = 0, j <= JMax[mMax, n0], j++,
      n = n0 + j*t;
      m = M[c, j];
      If[m <= mMax, AppendTo[result, {g0, g1, c, m, n}]];
      ];
     ];
    ];
   result
   ];
(* Multiplier list *)
MList[mMax_] := Union[MasterTable[mMax][[All, 4]]];

Crossrefs

Cf. A000045.

Sequence in context: A268962 A121104 A118627 * A080787 A065124 A048817

Adjacent sequences: A358401 A358402 A358403 * A358405 A358406 A358407

Keyword

nonn

Author

David Terr, Nov 14 2022

Status

approved