OFFSET
1,1
COMMENTS
A 4-tuple (m,u,v,w) is a (p,q)-quartal if m,u,v,w are positive integers such that m<v and m^p + u^q = v^p + w^q. Here, p=2, q=3, m=1.
Includes all squares > 1, as 1 + (i^2)^3 = v^2 + w^3 with w = 1, v = i^3. - Robert Israel, Jul 28 2025
LINKS
Robert Israel, Table of n, a(n) for n = 1..1100
EXAMPLE
First 20 (2,3)-quartals (1,u,v,w):
m u v w
1 4 8 1
1 9 27 1
1 12 27 10
1 16 64 1
1 25 125 1
1 27 134 12
1 29 123 21
1 32 181 2
1 35 126 30
1 35 207 3
1 36 216 1
1 40 251 10
1 41 253 17
1 42 217 30
1 42 269 12
1 47 300 24
1 48 267 34
1 49 343 1
1 51 242 42
1 54 379 24
1^2 + 12^3 = 27^2 + 10^3 = 1729, so (1,12,27,10) is in the list.
MAPLE
f:= proc(u) local t;
t:= 1+u^3;
u$nops(select(w -> issqr(t-w^3), [$1 .. u-1]))
end proc:
map(f, [$1..1000]); # Robert Israel, Jul 28 2025
MATHEMATICA
quart[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]}]; ]; ]; ];
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 = quart[1, 2, 3, 6000]
(* Peter J. C. Moses, Jun 21 2025 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 28 2025
STATUS
approved
