On Representation of an Integer by X 2 + Y 2 + Z 2 and the Modular Equations of Degree 3 and 5 (original) (raw)
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The Ramanujan Journal, 2009
We revisit old conjectures of Fermat and Euler regarding representation of integers by binary quadratic form x 2 + 5y 2 . Making use of Ramanujan's 1 ψ 1 summation formula we establish a new Lambert series identity for P ∞ n,m=−∞ q n 2 +5m 2 . Conjectures of Fermat and Euler are shown to follow easily from this new formula. But we don't stop there. Employing various formulas found in Ramanujan's notebooks and using a bit of ingenuity we obtain a collection of new Lambert series for certain infinite products associated with quadratic forms such as x 2 + 6y 2 , 2x 2 + 3y 2 , x 2 + 15y 2 , 3x 2 + 5y 2 , x 2 + 27y 2 , x 2 + 5(y 2 + z 2 + w 2 ), 5x 2 + y 2 + z 2 + w 2 . In the process, we find many new multiplicative eta-quotients and determine their coefficients.
On the number of representations of certain integers as sums of 11 or 13 squares
Journal of Number Theory, 2003
Let r k (n) denote the number of representations of an integer n as a sum of k squares. We prove that for odd primes p, r 11 (p 2) = 330 31 (p 9 + 1) − 22(−1) (p−1)/2 p 4 + 352 31 H(p), where H(p) is the coefficient of q p in the expansion of q ∞ j=1 (1 − (−q) j) 16 (1 − q 2j) 4 + 32q 2 ∞ j=1 (1 − q 2j) 28 (1 − (−q) j) 8. This result, together with the theory of modular forms of half integer weight is used to prove that r 11 (n) = r 11 (n) 2 9 λ/2 +9 − 1 2 9 − 1 p p 9 λp/2 +9 − 1 p 9 − 1 − p 4 −n p p 9 λp/2 − 1 p 9 − 1 , where n = 2 λ p p λp is the prime factorisation of n and n is the square-free part of n, in the case that n is of the form 8k + 7. The products here are taken over the odd primes p, and n p is the Legendre symbol. We also prove that for odd primes p, r 13 (p 2) = 4030 691 (p 11 + 1) − 26p 5 + 13936 691 τ (p), where τ (n) is Ramanujan's τ function, defined by q ∞ j=1 (1 − q j) 24 = ∞ n=1 τ (n)q n. A conjectured formula for r 2k+1 (p 2) is given, for general k and general odd primes p.
Sums of Squares and Sums of Triangular Numbers
Motivated by two results of Ramanujan, we give a family of 15 results and 4 related ones. Several have interesting interpretations in terms of the number of representations of an integer by a quadratic form λ 1 x 2 1 + · · · + λ m x 2 m , where λ 1 + · · · + λ m = 2, 4 or 8. We also give a new and simple combinatorial proof of the modular equation of order seven.