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Vladimir Pletser

Publications and source records attributed to Vladimir Pletser.

24 records · Page 2Linked to original sources
arXiv2014

Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers

Considering the problem of finding all the integer solutions of the sum of $M$ consecutive integer squares starting at $a^{2}$ being equal to a squared integer $s^{2}$, it is shown that this problem has no solutions if $M\equiv3,5,6,7,8$ or $10 (mod\,12)$ and has integer solutions if $M\equiv0,9,24$ or $33 (mod\,72)$; or $M\equiv1,2$ or $16 (mod\,24)$; or $M\equiv11 (mod\,12)$. All the allowed values of $M$ are characterized using necessary conditions. If $M$ is a square itself, then $M\equiv1 (mod\,24)$ and $(M-1)/24$ are all pentagonal numbers, except the first two.

math.NT↗
arXiv2014

Additional congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers

The problem of finding all the integer solutions in $a$, $M$ and $s$ of sums of $M$ consecutive integer squares starting at $a^{2}\geq1$ equal to squared integers $s^{2}$, has no solutions if $M\equiv3,5,6,7,8$ or $10\left(mod\,12\right)$ and has integer solutions if $M\equiv0,9,24% or %33\left(mod\,72\right)$; or $M\equiv1,2$ or $16\left(mod\,24\right)$; or $M\equiv11\left(mod\,12\right)$. In this paper, additional congruence conditions are demonstrated on the allowed values of $M$ that yield solutions to the problem by using Beeckmans' eight necessary conditions, refining further the possible values of $M$ for which the sums of $M$ consecutive integer squares equal integer squares.

math.HO↗
arXiv2013

Conjecture on the value of Pi(10^26), the number of primes less than 10^26

Based on the first 25 known values of Pi(10^n), the number of primes less than 10^n, with n integer between 1 and 25, we propose a conjectured value range of Pi(10^26) calculated by using polynomial interpolations with two corrective functions obtained by Thiele interpolations on relative differences of exact and interpolated values of Pi(10^n). The conjectured range value is in agreement with values obtained by the Eulerian logarithmic integral and with the Riemann functions.

math.NT↗
arXiv2012

Does the Ishango Bone Indicate Knowledge of the Base 12? An Interpretation of a Prehistoric Discovery, the First Mathematical Tool of Humankind

In the early fifties, the Belgian Prof. J. de Heinzelin discovered a bone in the region of a fishermen village called Ishango, at one of the farthest sources of the Nile, on the border of Congo and Uganda. The Heinzelin's Ishango bone has notches that seem to form patterns, making it the first tool on which some logic reasoning seems to have been done. In this paper a new interpretation is proposed for these patterned notches, based on a detailed observation of their structure. It can be called the "slide rule"-reading, in contrast to former "arithmetic game" and "calendar" explanations. Additional circumstantial evidences are given to support the hypothesis that the Ishango bone is a primitive mathematical tool using the base 12 and sub-bases 3 and 4.

math.HO↗