A217470 The Diophantine equation x^2 - x*y - G*y^2 = -1, G a positive integer, D = 4*G + 1 not a perfect square, has no solution precisely for G = a(n).

5, 8, 11, 14, 17, 19, 23, 26, 29, 32, 33, 35, 38, 40, 41, 44, 47, 50, 51, 52, 53, 54, 55, 59, 61, 62, 63, 65, 68, 71, 74, 75, 76, 77, 80, 82, 83, 85, 86, 89, 92, 94, 95, 96, 98, 101

Offset: 1

Author

Wolfdieter Lang, Oct 04 2012

Keywords

• nonn

Comments

See the Perron reference for the theorem which by negation implies that this quadratic Diophantine equation has no solution if and only if A077427 is even.

See the pairs (x, y) = (A077057, A077058) which for these a(n) values are the smallest positive solutions of the Diophantine equation x^2 - x*y - a(n)*y^ = +1.

In the table on p. 108 of the Perron reference these a(n) values, called there also G, are the ones were in the third column numbers in brackets appear.

The case D = 4*G + 1 = m^2 > 1 has trivially no solutions: the equation is then X^2 - Y^2 = -4, with X = |2*x-y|, Y = |m*y|. X and Y are either both even or both odd. In the first case one is led to the equation v^2 - w^2 = (v-w)*(v+w)= -1, with X = 2*v and Y = 2*w, and there is only the solution (v,w) = (0,1), hence 2*x = y, m*y = 2. But then m=2 and y=1 with non-integer x solution. In the other case X = 2*v+1 and Y = 2*w+1, v not w, leading to v + w + 1 = -1 with no positive integer solution. Thanks to T. D. Noe for pointing out that one has to mention that these values G = A002378(k), k>=1, with D a perfect (odd) square, are here not included.

Examples

			a(1) = 5 because 5 = A078358(4) and A077427(4) = 2, which is even.
		

References

• O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108).

Crossrefs

Cf. A077057, A077058, A078358, A077427.

Formula

a(n) gives the increasingly ordered values for G from A078358 which appear at position k where A077427(k) is even, for k>=1. The next even number in A077427 appears for k = 6 and

A078358(6) = 8, hence a(2) = 8.