arXiv subjects
Publications and source records attributed to Leonhard Euler.
Translation of the Latin original "Speculationes circa quasdam insignes proprietates numerorum" (1784). E564 in the Enestrom index. In this paper Euler talks about Farey sequences and proves some results about the phi function, the number of positive integers less than and relatively prime to an integer. Euler uses the notation pi instead of phi.
Translation of "Methodus succincta summas serierum infinitarum per formulas differentiales investigandi" (1780). Euler wants to represent some given series of functions S(x)=X(x)+X(x+1)+X(x+2)+etc. in a different way. He writes S as a series in derivatives of X with unknown coefficients. He makes a generating function V(z) out of these coefficients, which is the same as a generating function that involves the Bernoulli numbers.
This paper contains a number of letters exchanged between MM. Dollond, Short, and Euler in 1752 regarding the construction and use of various objective lenses. The final letters, authored by Euler, appear in translation for the first time. This is numbered E210 in Enestrom's index.
Euler states without proof statements about the form of prime divisors of numbers of the form aa+Nbb. See Ed Sandifer's How Euler Did It, ``Factors of Forms'', December 2005 at http://www.maa.org/news/howeulerdidit.html for a summary of the paper.
Euler wants to find rational numbers (integers) x and y such that x+y is a square and x^2+y^2 is a fourth power. He parametrizes these with two other variables that satisfy certain equations.
In this paper Euler shows how, if we have recursive functions f,g,h and an infinite sequence A,B,C,... which satisfies fA=gB+hC, f'B=g'C+h'D, f''C=g''D+h''E, f'''D=g'''E+h'''F, etc., where the primes denote an index not a derivative, then we can find a continued fraction for fA/B.
In this short paper Euler gives a highly convergent series for arctan and thus pi, which converges much faster than the Leibniz series for arctan.
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.
In this paper Euler shows that there are no additional square idoneal numbers aside from 1, 4, 9, 16, and 25.
In this paper, Euler gives the general trionomial coefficient as a sum of the binomial coefficients, the general quadrinomial coefficient as a sum of the binomial and trinomial coefficients, the general quintonomial coefficient as a sum of the binomial and quadrinomial coefficients, and gives a general determination of the coefficients of the expansion of any polynomial (1+x+x^2+...+x^m)^n as a sum of the coefficients of lower degree polynomial coefficients.
In this paper Euler considers the properties of the pentagonal numbers, those numbers of the form $\frac{3n^2 \pm n}{2}$. He recalls that the infinite product $(1-x)(1-x^2)(1-x^3)...$ expands into an infinite series with exponents the pentagonal numbers, and tries substituting the roots of this infinite product into this infinite series. I am not sure what he is doing in some parts: in particular, he does some complicated calculations about the roots of unity and sums of them, their squares, reciprocals, etc., and also sums some divergent series such as 1-1-1+1+1-1-1+1+..., and I would appreciate any suggestions or corrections about these parts.
This is a translation of Leonhard Euler's ``De quadratis magicis'' . It is E795 in the Enestrom index. This paper studies how to construct magic squares with certain numbers of cells, in particular 9, 16, 25 and 36. It considers some general rules for making squares of even and odd orders. Euler uses Graeco-Latin squares and constrains the values that the variables can take to make magic squares. I think this is where the terms ``Latin'' and ``Graeco-Latin'' square comes from, since he uses Latin letters in one square and Greek (i.e. ``Graeco'') letters in another for each construction. (This paper was published before Euler's only other paper ``Recherches sur une nouvelle espece de quarres magiques'' E530 about Latin squares was, in 1782.)
This is an English translation from the Latin original of Leonhard Euler's ``Solutio facilior problematis Diophantei circa triangulum, in quo rectae ex angulis latera opposita bisecantes rationaliter exprimantur''. In this paper, Euler proves that there exist triangles with integer length sides such that the length of the bisectors of the sides to the opposite angles are integer valued, and he gives a general method for making a certain class of such triangles.
This seems to be the first English translation of this paper from the French original, ``Sur les rentes viageres''. In the paper, Euler gives a general formula for calculating the price of a life annuity that yields a certain amount per year, assuming the annuity manager can get a 5 percent return, for people of different ages. He also gives formulas to calculate the price of annuities that only start to pay out a certain number of years after they are purchased. He gives many numerical examples, giving tables for the prices of annuities for annuitants up to 90 years old.
This is a translation into English from the original Latin of Leonhard Euler's Exercitatio analytica, Nova Acta Academiae Scientarum Imperialis Petropolitinae 8 (1794), 69-72; E664 in the Enestrom index. In it Euler uses the infinite product identity for sin to prove some properties of the series for the Riemann zeta function for even n.