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Full-Text Articles in Physical Sciences and Mathematics
Supplement To "On Translations Of Quadratic Residues", Bruce Brandt
Supplement To "On Translations Of Quadratic Residues", Bruce Brandt
Journal of the Minnesota Academy of Science
Let n and k be arbitrary positive integers. We will tend to be concerned with small k and with n which are several times k!. I stated two conditions on (n, k) in a previous paper; in this paper I restate them and further explore them. In particular, it is proven that if n is the least number satisfying Condition 1 for a certain k, then the least number for k + l must be at least 2n + l. Condition 1 and Condition 2 are rephrased graph-theoretically. A heuristic explanation for why the quadratic. residues tend to satisfy Condition …
More Triangular Number Results, Bruce Brandt
More Triangular Number Results, Bruce Brandt
Journal of the Minnesota Academy of Science
I define an increasing function from triangular numbers to triangular numbers and prove it preserves [mathematical symbol]. I conjecture that whether a triangular number is in the image of this function is related to the magnitude of [mathematical symbol] on the triangular number. Parallel theorems and conjectures exist for pentagonal numbers. I also make conjectures about the partial sums of [mathematical symbol] on the triangular numbers along with a conjecture about the sums of absolute values of [mathematical symbol] on the squares.