A233044 Pairs p, q for those partial sums p/q of the series e = sum_{n>=0} 1/n! that are not convergents to e.

1, 1, 5, 2, 65, 24, 163, 60, 1957, 720, 685, 252, 109601, 40320, 98641, 36288, 9864101, 3628800, 13563139, 4989600, 260412269, 95800320, 8463398743, 3113510400, 47395032961, 17435658240, 888656868019, 326918592000

Offset: 1

Author

Jonathan Sondow, Dec 07 2013

Keywords

• nonn

Comments

Sondow (2006) conjectured that 2/1 and 8/3 are the only partial sums of the Taylor series for e that are also convergents to the simple continued fraction for e. Sondow and Schalm (2008, 2010) proved partial results toward the conjecture. Berndt, Kim, and Zaharescu (2012) proved it in full.

Examples

			1/1, 5/2, 65/24, 163/60, 1957/720, 685/252, 109601/40320, 98641/36288, 9864101/3628800, 13563139/4989600, 260412269/95800320, 8463398743/3113510400, 47395032961/17435658240, 888656868019/326918592000
		

References

• J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e?, (and a link to the primes 2, 5, 13, 37, 463), part I, in Tapas in Experimental Mathematics, T. Amdeberhan and V. H. Moll, eds., Contemp. Math., vol. 457, American Mathematical Society, Providence, RI, 2008, pp. 273-284.

Links

• B. Berndt, S. Kim, and A. Zaharescu, Diophantine approximation of the exponential function and Sondow's conjecture, abstract 2012.
• J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly, 113 (2006), 637-641.
• J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e?, (and a link to the primes 2, 5, 13, 37, 463), part II, in Gems in Experimental Mathematics, T. Amdeberhan, L. A. Medina, V. H. Moll, eds., Contemp. Math., vol. 517, American Mathematical Society, Providence, RI, 2010, pp. 349-363.

Crossrefs

Cf. A061354, A061355, A007676, A007677.

Formula

a(2n-1)/a(2n) = A061354(k)/A061355(k) for some k <> 1 and 3.

a(2n-1)/a(2n) <> A007676(k)/A007677(k) for all k.