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

Author

David Terr, Nov 14 2022

Keywords

• nonn
• more

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.

Examples

			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.
		

Crossrefs

Cf. A000045.

Programs

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]]];

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.