A178612 Positive numbers of the form p^6 - 4*p^4*q + 4*p^2*q^2 + 4*q^3 (and p*q <> 0).

5, 20, 32, 41, 124, 133, 140, 160, 189, 224, 257, 265, 284, 292, 305, 320, 445, 509, 581, 644, 673, 945, 985, 1076, 1085, 1120, 1280, 1345, 1436, 1489, 1541, 1597, 1708, 1772, 1917, 2048, 2237, 2273, 2336, 2345, 2489, 2624, 2749, 2889, 2980, 3105, 3140, 3205

Offset: 1

Author

Artur Jasinski, May 30 2010

Keywords

• nonn
• changed

Comments

Conjecture: There are no perfect squares in this sequence (in spite of all numbers being congruent to 0 or 1 mod 4).

If any perfect square occurred in this sequence then a septic trinomial x^7 + A*x^2 + B with two irreducible factors of degree 3 and 4 would exist.

This sequence is a subsequence of A079896.

Links

• Robin Visser, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A079896.

Programs

Magma

is_A178612 := function(k)
    R := PolynomialRing(Integers());
    for s in Solutions(Thue(x^3 - 4*x^2 + 4*x + 4), k) do
        if (s[1]*s[2]) ne 0 and IsSquare(s[1]) then return true; end if;
    end for;
    return false;
end function;
[k : k in [1..1000] | is_A178612(k)];  // Robin Visser, Aug 26 2025

Mathematica

aa = {}; Do[Do[kk = p^6 - 4 p^4 q + 4 p^2 q^2 + 4 q^3; If[(kk > 0) && (p q != 0), AppendTo[aa, kk]], {p, 1, 200}], {q, -200, 200}]; Take[Union[aa], 100]