Ramanujan's ternary quadratic form (original) (raw)
Related papers
On ternary quadratic forms over the rational numbers
2021
In this note, we give an elementary proof of the following classical fact. Any positive definite ternary quadratic form over the rational numbers fails to represent infinitely many positive integers. For any ternary quadratic form (positive definite or indefinite), our method constructs certain congruence classes whose elements, up to a square factor, are the only elements not represented over the rational numbers by that form. In the case of a positive definite ternary form, we show that these classes are non-empty. This shows that the minimum number of variables in a positive definite quadratic form representing all positive integers is four. Our proof is very elementary and only uses quadratic reciprocity of Gauss.
Essentially Unique Representations by Certain Ternary Quadratic Forms
Experimental Mathematics, 2015
In this paper we generalize the idea of "essentially unique" representations by ternary quadratic forms. We employ the Siegel formula, along with the complete classification of imaginary quadratic fields of class number less than or equal to 8, to deduce the set of integers which are represented in essentially one way by a given form which is alone in its genus. We consider a variety of forms which illustrate how this method applies to any of the 794 ternary quadratic forms which are alone in their genus. As a consequence, we resolve some conjectures of Kaplansky regarding unique representation by the forms x 2 + y 2 + 3z 2 , x 2 + 3y 2 + 3z 2 , and x 2 + 2y 2 + 3z 2 [18].
On Integer Solutions of the Ternary Quadratic Equation
International Journal for Research in Applied Science & Engineering Technology (IJRASET), 2023
We derive distinct integral solutions in four different patterns. There are a few intriguing connections between the solutions and unique polygonal numbers that are presented.
Integral Positive Ternary Quadratic Forms
Developments in Mathematics, 2013
We discuss some families of integral positive ternary quadratic forms. Our main example is f (x, y, z) = x 2 + y 2 + 16nz 2 , where n is positive, squarefree, and n = u 2 + v 2 with u, v ∈ Z.
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.