A Square as the Sum of 7 Squares

Sandham, H. F.

doi:10.1093/qmath/4.1.230

Cited by 17 publications

(13 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(2.6) and (2.7) seem to have been first explicitly stated by Hurwitz [17] and [16], respectively, and (2.8) is due to Sandham [27]. For sums of nine squares we have…”

Section: Article In Pressmentioning

confidence: 78%

On the number of representations of certain integers as sums of 11 or 13 squares

Cooper

2003

Journal of Number Theory

View full text Add to dashboard Cite

Let r k ðnÞ denote the number of representations of an integer n as a sum of k squares. We prove that for odd primes p; r 11 ð p 2 Þ ¼ 330 31 ð p 9 þ 1Þ À 22ðÀ1Þ ð pÀ1Þ=2 p 4 þ 352 31 Hð pÞ;where Hð pÞ is the coefficient of q p in the expansion ofThis result, together with the theory of modular forms of half integer weight is used to prove that r 11 ðnÞ ¼ r 11 ðn 0 Þ 2 9Il=2mþ9 À 1 2 9 À 1

show abstract

“…(2.6) and (2.7) seem to have been first explicitly stated by Hurwitz [17] and [16], respectively, and (2.8) is due to Sandham [27]. For sums of nine squares we have…”

Section: Article In Pressmentioning

confidence: 78%

On the number of representations of certain integers as sums of 11 or 13 squares

Cooper

2003

Journal of Number Theory

View full text Add to dashboard Cite

show abstract

“…In this section we shall use our method to give explicit formulae for r 2k+1 (n 2 ) and r 2k+1 (p 2 ) for a few values of k. When 1 ≤ k ≤ 6, the previous expressions obtained in [14,15] can be derived using our method. We note that when k = 5, a(2n) = 16g(2n), where g(n) is the coefficient appearing in Sandham's formula [15], p. 32.…”

Section: Explicit Formulaementioning

confidence: 99%

“…For example, the formulas for r k (n) or r k (n 2 ) are known for k = 1, 3,5,7,9,11,13. See [2,4,5,7,8,[14][15][16] for details. These formulas were obtained either by an elementary method or by using the theory of modular forms.…”

mentioning

confidence: 99%

On the representation of integers as sums of an odd number of squares

Gun

Balasubramaniam

2008

Ramanujan J

View full text Add to dashboard Cite

In a recent work, S. Cooper (J. Number Theory 103:135-162, 1988) conjectured a formula for r 2k+1 (p 2 ), the number of ways p 2 can be expressed as a sum of 2k + 1 squares. Inspired by this conjecture, we obtain an explicit formula for r 2k+1 (n 2 ), n ≥ 1.A classical problem in number theory is to give an explicit formula for the number of ways one can represent a non-negative integer n as a sum of k squares, where k is a positive integer. The study of r k (n) has a long history. For k = 2, 4, 6, 8, elegant formulae for r k (n) were found by Jacobi. A general formula for r k (n), when k is even was stated by Ramanujan [12]. It was proved by Mordell [10] and S. Cooper [3] gave an elementary proof using Ramanujan's 1 ψ 1 summation formula. It is also known that r k (n) can be expressed in terms of coefficients of Eisenstein series and cusp forms and Rankin [13] showed that the cusp form part is non-trivial for k > 8. Recently, combining a variety of methods, S. Milne [9] obtained formulas for r 4s 2 (n) and r 4s 2 +4s (n) Dedicated to Srinivasa Ramanujan.S. Gun · B. Ramakrishnan ( )

show abstract

“…Various special cases of (3.1)-(3.7) have been given by many authors. For example, (this is by no means a complete list) see Cohen [2], Hurwitz [23,24], Mordell [35], Olds [36][37][38], Sandham [43] and Stieltjes [50,51]. For some additional results on sums of an odd number of squares, see Bateman [1], Hardy [19,20], Lomadze [30,31] and Sandham [44].…”

Section: Case 2 N ≡ 0 (Mod 4)mentioning

confidence: 99%