Concerning life annuities

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.Comment: 9 pages, English translation of ``Sur les rentes viageres'', Memoires de l'academie des sciences de Berlin 16 (1767), 165-17

A double demonstration of a theorem of Newton, which gives a relation between the coefficient of an algebraic equation and the sums of the powers of its roots

Translation from the Latin original, "Demonstratio gemina theorematis Neutoniani, quo traditur relatio inter coefficientes cuiusvis aequationis algebraicae et summas potestatum radicum eiusdem" (1747). E153 in the Enestrom index. In this paper Euler gives two proofs of Newton's identities, which express the sums of powers of the roots of a polynomial in terms of its coefficients. The first proof takes the derivative of a logarithm. The second proof uses induction and the fact that in a polynomial of degree $n$, the coefficient of $x^{n-k}$ is equal to the sum of the products of $k$ roots, times $(-1)^k$.Comment: 9 page

A solution to a problem of Fermat, on two numbers of which the sum is a square and the sum of their squares is a biquadrate, inspired by the Illustrious La Grange

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.Comment: ``Solutio problematis Fermatiani de duobus numeris, quorum summa sit quadratum, quadratorum vero summa biquadratum, ad mentem illustris La Grange adornata'', Memoires de l'Academie Imperiale des Sciences de St.-Petersbourg 10 (1826), 3-6. E769 in the Enestrom inde

Speculations on some characteristic properties of numbers

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.Comment: 9 page

Theorems about the divisors of numbers contained in the form paa±qbbpaa \pm qbb

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.Comment: 24 pages, translation of ``Theoremata circa divisores numerorum in hac forma $paa \pm qbb$ contentorum'', Commentarii academiae scientiarum Petropolitanae 14 (1751), 151-18

An easier solution of a Diophantine problem about triangles, in which those lines from the vertices which bisect the opposite sides may be expressed rationally

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.Comment: 6 pages, seems to be first English translation of Euler's ``Solutio facilior problematis Diophantei circa triangulum, in quo rectae ex angulis latera opposita bisecantes rationaliter exprimantur'', Memoires de l'Academie Imperiale des Sciences de St.-Petersbourg 2 (1810), 10-1

A succinct method for investigating the sums of infinite series through differential formulae

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.Comment: 8 page

Theorems on residues obtained by the division of powers

This is an English translation of Euler's ``Theoremata circa residua ex divisione potestatum relicta'', Novi Commentarii academiae scientiarum Petropolitanae 7 (1761), 49-82. E262 in the Enestrom index. Euler gives many elementary results on power residues modulo a prime number p. He shows that the order of a subgroup generated by an element a in F_p^* must divide the order p-1 of F_p^* (i.e. a special case of Lagrange's theorem for cyclic groups). Euler also gives a proof of Fermat's little theorem, that a^{p-1} = 1 mod p for a relatively prime to p (i.e. not 0 mod p). He remarks that this proof is more natural, as it uses multiplicative properties of F_p^* instead of the binomial expansion. Thanks to Jean-Marie Bois for pointing out some typos.Comment: 27 pages, E26

Finding the sum of any series from a given general term

Translation from the Latin original, "Inventio summae cuiusque seriei ex dato termino generali" (1735). E47 in the Enestrom index. In this paper Euler derives the Euler-Maclaurin summation formula, by expressing y(x-1) with the Taylor expansion of y about x. In sections 21 to 23 Euler uses the formula to find expressions for the sums of the nth powers of the first x integers. He gives the general formula for this, and works it out explicitly up to n=16. In sections 25 to 28 he applies the summation formula to getting approximations to partial sums of the harmonic series, and in sections 29 to 30 to partial sums of the reciprocals of the odd positive integers. In sections 31 to 32, Euler gets an approximation to zeta(2); in section 33, approximations for zeta(3) and zeta(4). I found David Pengelley's paper "Dances between continuous and discrete: Euler's summation formula", in the MAA's "Euler at 300: An Appreciation", edited by Robert E. Bradley, Lawrence A. D'Antonio, and C. Edward Sandifer, very helpful and I recommend it if you want to understand the summation formula better.Comment: 13 page

On the remarkable properties of the pentagonal numbers

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.Comment: 16 pages, seems to be first English translation of Euler's Latin original ``De mirabilis proprietatibus numerorum pentagonalium'', Acta Academiae Scientarum Imperialis Petropolitinae 4 (1783), no. 1, 56-75. E54