A386219 Values of v in the (1,3)-quartals (m,u,v,w) having m=4; i.e., values of v for solutions to 4 + u^3 = v + w^3, in positive integers, with m

11, 23, 30, 41, 60, 67, 65, 102, 121, 128, 95, 156, 193, 212, 219, 131, 222, 283, 320, 339, 346, 173, 300, 391, 452, 489, 508, 515, 221, 390, 517, 608, 669, 706, 725, 732, 275, 492, 661, 788, 879, 940, 977, 996, 1003, 335, 606, 823, 992, 1119, 1210, 1271, 1308

Offset: 1

Author

Clark Kimberling, Jul 28 2025

Keywords

• nonn

Comments

A 4-tuple (m,u,v,w) is a (p,q)-quartal if m,u,v,w are positive integers such that m

Examples

			First 20 (1,3)-quartals (4,u,v,w):
  m   u    v   w
  4   2   11   1
  4   3   23   2
  4   3   30   1
  4   4   41   3
  4   4   60   2
  4   4   67   1
  4   5   65   4
  4   5  102   3
  4   5  121   2
  4   5  128   1
  4   6   95   5
  4   6  156   4
  4   6  193   3
  4   6  212   2
  4   6  219   1
  4   7  131   6
  4   7  222   5
  4   7  283   4
  4   7  320   3
  4   7  339   2
4^1 + 5^3 = 65^1 + 4^3, so (4,5,65,4) is in the list.
		

Crossrefs

Cf. A385882, A386215, A386217.

Programs

Mathematica

quartals[m_, p_, q_, max_] := Module[{ans = {}, lhsD = <||>, lhs, v, u, w, rhs},
   For[u = 1, u <= max, u++, lhs = m^p + u^q;
    AssociateTo[lhsD, lhs -> Append[Lookup[lhsD, lhs, {}], u]];];
   For[v = m + 1, v <= max, v++,
    For[w = 1, w <= max, w++, rhs = v^p + w^q;
      If[KeyExistsQ[lhsD, rhs],
       Do[AppendTo[ans, {m, u, v, w}], {u, lhsD[rhs]}];];];];
   ans = SortBy[ans, #[[2]] &];
   Do[Print["Solution ", i, ": ", ans[[i]], " (", m, "^", p, " + ",
     ans[[i, 2]], "^", q, " = ", ans[[i, 3]], "^", p, " + ",
     ans[[i, 4]], "^", q, " = ", m^p + ans[[i, 2]]^q, ")"], {i,
     Length[ans]}]; ans];
solns = quartals[4, 1, 3, 2000]
Grid[solns]
(* Peter J. C. Moses, Jun 20 2025 *)

Extensions

Data corrected by Sean A. Irvine, Aug 11 2025

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