A213687 Numbers which are the values of the quadratic polynomial 3+4*k+7*t+8*k*t on nonnegative integers.

3, 7, 10, 11, 15, 17, 19, 22, 23, 24, 27, 31, 34, 35, 37, 38, 39, 43, 45, 46, 47, 51, 52, 55, 57, 58, 59, 63, 66, 67, 70, 71, 73, 75, 77, 79, 80, 82, 83, 87, 91, 94, 95, 97, 99, 101, 103, 106, 107, 108, 111, 112, 115, 117, 118, 119, 122, 123, 126, 127, 129

Offset: 1

Author

Michel Mizony, Jun 18 2012

Keywords

• nonn

Comments

For all these numbers a(n) we have the following Erdos-Straus decomposition: 4/p = 4/(3+4*k+7*t+8*k*t) = 1/(2*(3+4*k+7*t+8*k*t)*(1+k)) + 1/((1+k)*(2*t+1)) + 1/(2*(1+k)*(2*t+1)*(3+4*k+7*t+8*k*t));

Moreover this sequence is related to irreducible twin Pythagorean triples: the associated Pythagorean triple is [2t(t+1),2t+1, 2t(t+1)+1].

In 1948 Erdos and Straus conjectured that for any positive integer n >= 2 the equation 4/n = 1/x + 1/y +1/z has a solution with positive integers x, y and z (without the additional requirement 0 < x < y < z).

For the solution (x,y,z) having the largest z value, see (A075245, A075246, A075247).

Examples

			31 is a term because the solutions to 3+4*k+7*t+8*k*t = 31 are {k = 0, t = 4}, {k = 7, t = 0}.
		

References

• I. Gueye and M. Mizony, Recent progress about Erdős-Straus conjecture, B SO MA S S, Volume 1, Issue 2, pp. 6-14.
• M. Mizony and I. Gueye, Towards the proof of Erdős-Straus conjecture, B SO MA S S, Volume 1, Issue 2, pp. 141-150.

Links

• P. Erdős, On a Diophantine equation, (Hungarian. Russian, English summaries), Mat. Lapok 1, 1950, pp. 192-210.
• M. Mizony and M.-L. Gardes, Sur la conjecture d'Erdős et Straus, see pages 14-17.
• Eric Weisstein's World of Mathematics, Twin Pythagorean Triple
• K. Yamamoto, On the Diophantine Equation 4/n=1/x+1/y+1/z, Mem. Fac. Sci. Kyushu U. Ser. A 19, 37-47, 1965.

Crossrefs

Cf. A213340 (the quadratic polynomial 5+8t+12k+16kt).

Cf. A001844 (centered square numbers: 2n(n+1)+1).

Cf. A005408 (x values), A046092 (y values).

Cf. A073101 (number of solutions (x,y,z) to 4/n = 1/x + 1/y + 1/z satisfying 0 < x < y < z).

Programs

Maple

H:=(k, t) -> 4/(3+4*k+7*t+8*k*t) = [1/2*1/((3+4*k+7*t+8*k*t)*(1+k)), 1/((1+k)*(2*t+1)), 1/2*1/((1+k)*(2*t+1)*(3+4*k+7*t+8*k*t))]:
cousin:=proc(p)
local n,k;
for n from 0 to (p-3)/7 do
if (p-3-7*n) mod (4+8*n)=0  then k:=(p-3-7*n)/(4+8*n):
return([p,n,H(k,n)]) fi;od;
end:
L:=NULL:for p from 2 to 500 do L:=L,cousin(p): od:{L}[1..10];map(u->op(1,u),{L});map(u->op(2,u),{L});