arXiv subjects

Artur Diener

Publications and source records attributed to Artur Diener.

16 recordsLinked to original sources

arXiv2012

Theoremata arithmetica nova methodo demonstrata

Euler presents a third proof of the Fermat theorem, the one that lets us call it the Euler-Fermat theorem. This seems to be the proof that Euler likes best. He also proves that the smallest power x^n that, when divided by a numer N, prime to x, and that leaves a remainder of 1, is equal to the number of parts of N that are prime to n, that is to say, the number of distinct aliquot parts of N. The translation is presnted from Euler's Latin original into German.

math.HO↗

arXiv2012

Theorematum quorundam arithmeticorum demonstrationes

Euler proves that the sum of two 4th powers can't be a 4th power and that the difference of two distinct non-zero 4th powers can't be a 4th power and Fermat's theorem that the equation x(x+1)/2=y^4 can only be solved in integers if x=1 and the final theorem y^3+1=x^2 can only be solves for x=3 and y=2 in integers. The paper is translated from Euler's Latin original into German.

math.HO↗

arXiv2012

De seriebus divergentibus

Euler gives a long introduction, giving all the arguments for and against the use of divergent series in calculus and then gives his own definition of the sum of a diverging series. Then in the second half of this paper he evaluates the the 1-1+2-6+24-120+720-... on several ways and gets the sum 0.5963473621372. The paper is translated from Euler's Latin original into German.

math.HO↗

arXiv2012

Evolutio producti infiniti (1-x)(1-xx)(1-x^3)(1-x^4)(1-x^5) etc. in seriem simplicem

This paper does exactly what the title says it does. It expands the given series to arrive at the familiar "pentagonal number" expansion, also known as the pentagonal number theorem, and recalls its application to partition numbers. The paper is translated from Euler's Latin original into German.

math.HO↗

arXiv2012

Nova series infinita maxime convergens perimetrum ellipsis exprimens

The title says what is done here. Euler finds a news series for the arc of an ellipse. The paper is translated from the Latin original into German.

math.HO↗

arXiv2012

De transformatione seriei divergentis 1 - mx + m(m+n)x^2 - m(m+n)(m+2n)x^3 + etc. in fractionem continuam

In this paper, Euler transforms the divergent series in the title, and thereby dervies the well known continued fraction expansion for pi/4 from Leibniz's series. The paper is translated from Euler's Latin originial into German.

math.HO↗

arXiv2012

Nova demonstratio, quod evolutio potestatum binomii Newtoniana etiam pro exponentibus fractis valeat

Here Euler notes the recursive relation for the general binomial coefficients, by assuming that (1+x)^a can be expanded in a power series.

math.HO↗

arXiv2012

Observationes generales circa series, quarum termini secundum sinus vel cosinus angulorum multiplorum progrediuntur

This paper, along with E592 and E636, seems to consider the binomial expansion (1+z)^n in the case where z is complex. Euler even gives the sums of divergent series. The paper is translated from Euler's Latin original into German.

math.HO↗

arXiv2012

Variae considerationes circa series hypergeometricas

Euler gives an asymptotic approximation for the function f(x) and recognizes that he is trying to interpolate the factorial function introduced in E19 "De progressionibus transcendentibus seu quarum termini generales algebraice dari nequeunt". The paper is translated from Euler's Latin original into German.

math.HO↗

arXiv2012

Disquitiones analyticae super evolutione potestatis trinomialis (1+x+xx)^n

Euler investigates the Taylorseries of (1+x+xx)^n and uses the results to evaluates some integrals which are today often proved with the calculus of residues.

math.HO↗

arXiv2012

Methodus succincta summas serierum infinitarum per formulas differentiales investigandi

This paper has more formal series with Bernoulli numbers and the Riemann zeta function. He gives the generating function for the Bernoulli numbers. The paper is translated from the Latin original into German.

math.HO↗

arXiv2012

Commentatio in fractionem continuam, qua illustris La Grange potestates binomiales expressit

Euler gives a continued fraction representation of (1 + x)n. involving 1,3,5,7,... and n^2-1,n^2-4,n^3-9,... and squares of z, for x=2y and y=z/(1-z). He evaluates this continued fraction at z=t sqrt(-1), for "vanishing" n, and for infinite n and deduces a continued fraction for log, arctan, etc. The paper is translated from Euler's Latin original into German.

math.HO↗

arXiv2012

Speculationes super formula integrali {\int} (x^ndx)/{\surd}(aa-2bx+cxx), ubi simul egregiae observationes circa fractiones continuas occurrunt

Euler evaluates the integrals in the title and recognizes a recursion between them, which he then uses to give continued fractions for the log and arctan. The paper is translated from Euler's Latin original into German.

math.HO↗

arXiv2012

Demonstratio insignis theorematis numerici circa uncias potestatum binomialium

This paper is about the product z^q/(1 - z)^(q + 1)(1 + (z/(1 - z)))^p, Euler gives the Talylor-Series and takes a closer look at the coefficient.

math.HO↗

arXiv2012

De termino generali serierum hypergeometricarum

Euler defines a function f(x) somehow as an infinite product and a generalization of [x], where [x] ist, what we now call following Legendre the Gamma-Funktion. He gets some recursive relationships for f(x), by applying some very nice tricks and using the asymptotics of the infinite products. The paper is translated from Latin into German.

math.HO↗

arXiv2012

Specimen transformationis singularis serierum

Euler starts with a hypergeometric series F(a, b, c, x), and differentiates it to get a functional relation. This relation is today known as Euler's identity. Then he integrates to get another and ends up with something like Legendre polynomials. The paper is translated from Euler's Latin original into German.

math.HO↗