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Schmidt (1993) proposed the problem of determining if for any integer , the sequence of numbers defined by the binomial sums(1)are all integers.The following table gives the first few values of for small .OEISvalues1A0018501, 3, 13, 63, 321, 1683, 8989, 48639, ...2A0052591, 5, 73, 1445, 33001, 819005, ...3A0928131, 9, 433, 36729, 3824001, 450954009, ...4A0928141, 17, 2593, 990737, 473940001, ...5A0928151, 33, 15553, 27748833, 61371200001, ...This was proved by Strehl (1993, 1994) and Schmidt (1995) for the case , corresponding to the Franel numbers. Strehl (1994) also found an explicit expression for the case . The resulting identities for are therefore known as the Strehl identities. The problem was restated in Graham et al. (1994, pp. 256 and 549), who indicated that H. S. Wilf had shown to be an integer for any for (Zudilin 2004).The problem was answered in the affirmative by Zudilin (2004), who found explicit expressions..

where is an Eulerian number and is a binomial coefficient (Worpitzky 1883; Comtet 1974, p. 242).

In his Meditationes algebraicae, Waring (1770, 1782) proposed a generalization of Lagrange's four-square theorem, stating that every rational integer is the sum of a fixed number of th powers of positive integers, where is any given positive integer and depends only on . Waring originally speculated that , , and . In 1909, Hilbert proved the general conjecture using an identity in 25-fold multiple integrals (Rademacher and Toeplitz 1957, pp. 52-61).In Lagrange's four-square theorem, Lagrange proved that , where 4 may be reduced to 3 except for numbers of the form (as proved by Legendre; Hardy 1999, p. 12). In 1909, Wieferich proved that . In 1859, Liouville proved (using Lagrange's four-square theorem and Liouville polynomial identity) that . Hardy, and Little established , and this was subsequently reduced to by Balasubramanian et al. (1986). For the case , in 1896, Maillet began with a proof that , in 1909 Wieferich proved , and..

Closed forms are known for the sums of reciprocals of even-indexed Lucasnumbers(1)(2)(3)(4)(5)(OEIS A153415), where is the golden ratio, is a q-polygamma function, and is a Jacobi theta function, and odd-indexed Lucas numbers(6)(7)(8)(9)(10)(11)(OEIS A153416), where is a Lambert series (Borwein and Borwein 1987, pp. 91-92). This gives the reciprocal Lucas constant as(12)(13)(14)(15)(16)(OEIS A093540), where is the golden ratio and is a Fibonacci number.Borwein and Borwein (1987, pp. 94-101) give a number of related beautiful formulas.

Closed forms are known for the sums of reciprocals of even-indexed Fibonaccinumbers(1)(2)(3)(4)(5)(6)(7)(OEIS A153386; Knopp 1990, Ch. 8, Ex. 114; Paszkowski 1997; Horadam 1988; Finch 2003, p. 358; E. Weisstein, Jan. 1, 2009; Arndt 2012), where is the golden ratio, is a q-polygamma function, and is a Lambert series (Borwein and Borwein 1987, pp. 91 and 95) and odd-indexed Fibonacci numbers(8)(9)(10)(11)(12)(13)(OEIS A153387; Landau 1899; Borwein and Borwein 1997, p. 94; E. Weisstein, Jan. 1, 2009; Arndt 2012), where is a Jacobi elliptic function. Together, these give a closed form for the reciprocal Fibonacci constant of(14)(15)(16)(17)(18)(OEIS A079586; Horadam 1988; Griffin 1992; Zhao 1999; Finch 2003, p. 358). The question of the irrationality of was formally raised by Paul Erdős and this sum was proved to be irrational by André-Jeannin (1989).Borwein..

The first Strehl identity is the binomial sum identity(Strehl 1993, 1994; Koepf 1998, p. 55), which are the so-called Franel numbers. For , 2, ..., the first few terms are 1, 2, 10, 56, 346, 2252, 15184, 104960, ... (OEIS A000172).The second Strehl identity is the binomial sum identity(Strehl 1993, 1994; Koepf 1998, p. 55) that is the case of Schmidt's problem. For , 1, 2, ..., these give the Apéry numbers 1, 5, 73, 1445, 33001, 819005, ... (OEIS A005259).

The important binomial theorem states that(1)Consider sums of powers of binomial coefficients(2)(3)where is a generalized hypergeometric function. When they exist, the recurrence equations that give solutions to these equations can be generated quickly using Zeilberger's algorithm.For , the closed-form solution is given by(4)i.e., the powers of two. obeys the recurrence relation(5)For , the closed-form solution is given by(6)i.e., the central binomial coefficients. obeys the recurrence relation(7)Franel (1894, 1895) was the first to obtain recurrences for ,(8)(Riordan 1980, p. 193; Barrucand 1975; Cusick 1989; Jin and Dickinson 2000), so are sometimes called Franel numbers. The sequence for cannot be expressed as a fixed number of hypergeometric terms (Petkovšek et al. 1996, p. 160), and therefore has no closed-form hypergeometric expression.Franel (1894, 1895) was also the first to obtain the recurrence..

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