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Proofs of Chappelon and Alfons\'ın Conjectures On Square Frobenius Numbers and its Relationship to Simultaneous Pell's Equations
Damanvir Singh Binner
Recently, Chappelon and Alfons\'ın defined the square Frobenius number of coprime numbers and to be the largest perfect square that cannot be expressed in the form …
A Complete List of all Numbers not of the Form
Damanvir Singh Binner
For given coprime positive integers and , the classical Frobenius coin problem asked to find the largest number that cannot be expressed in the form for nonnegative…
Reciprocity Relations for Summations of Squares of Floor Functions and Fractional Parts of Fractions
Damanvir Singh Binner
Given positive coprime integers and and a natural number , we obtain reciprocity relations which can be used to quickly evaluate summations like $\sum_{i=1}^{h} \{\frac{…
Some Bounds for Number of Solutions to and their Applications
Damanvir Singh Binner
In a recent work, the present author developed an efficient method to find the number of solutions of in non-negative integer triples where and a…
Generalization of a Result of Sylvester Regarding the Frobenius Coin Problem and an Elementary Proof of Eisenstein's Lemma for Jacobi Symbols
Damanvir Singh Binner
In a recent work, the present author generalized a fundamental result of Gauss related to quadratic reciprocity, and also showed that the above result of Gauss is equivalent to a s…