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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.
Some Conjectures Concerning Triangular Numbers, Bruce Brandt
Some Conjectures Concerning Triangular Numbers, Bruce Brandt
Journal of the Minnesota Academy of Science
Strong empirical evidence supports conjectures that certain number patterns always hold. These patterns concern the function cr, defined by the equation cr(n) = n - m2, m2 being the nearest square to n, on the domain of the triangular numbers. Triangular squares or triangular numbers of the form m2+m are also mentioned in most of the conjectures. One of the conjectures, for example, is that the sum of cr over the triangular numbers up to a triangular square is 0. Some of these patterns can be described by strings of symbols, such as "S" and "L," formed by first writing …
Supplement To "Some Conjectures Concerning Triangular Numbers", Bruce Brandt
Supplement To "Some Conjectures Concerning Triangular Numbers", Bruce Brandt
Journal of the Minnesota Academy of Science
In a previous paper (1), I stated many conjectures about triangular numbers. Since submitting that paper I have discovered many more results, including generalizations, which are presented here.