Searcharxiv⌕ Search

arXiv subjects

Leonhard Euler

Publications and source records attributed to Leonhard Euler.

At least 55 records · Page 3Linked to original sources
arXiv2008

An easy method for finding many very large prime numbers

Translated from the Latin original "Facillima methodus plurimos numeros primos praemagnos inveniendi" (1778). E718 in the Enestrom index. If m is a number of the form 4k+1 and is a sum of two relatively prime squares, then it is prime if and only if it is not the sum of two other squares. In this paper Euler wants to generalize this criterion. He looks at when 232aa+1 represents a prime number. He uses his "numeri idonei" (suitable numbers, convenient numbers, idoneal numbers).

math.HO↗
arXiv2008

On the values of integrals with the variable taken from $x=0$ to $x=\infty$

This is a translation from the Latin original, "De valoribus integralium a termino variabilis x=0 usque ad x=infinity extensorum" (1781). This is E675 in the Enestrom index. Euler wants to find the location of the end point of a clothoid, a type of spiral. He proves some general results about the gamma function.

math.HO↗
arXiv2008

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.

math.HO↗
arXiv2008

A conjecture on the forms of the roots of equations

E30 in the Enestrom index. Translated from the Latin original "De formis radicum aequationum cuiusque ordinis coniectatio" (1733). For an equation of degree n, Euler wants to define a "resolvent equation" of degree n-1 whose roots are related to the roots of the original equation. Thus by solving the resolvent we can solve the original equation. In sections 2 to 7 he works this out for quadratic, cubic and biquadratic equations. Apparently he gives a new method for solving the quartic in section 5. Then in section 8 Euler says that he wants to try the same approach for solving the quintic equation and general nth degree equations. In the rest of the paper Euler tries to figure out in what cases resolvents will work. Two references I found useful were Chapter 14, p.p. 106-113 of C. Edward Sandifer, "The Early Mathematics of Leonhard Euler", published 2007 by The Mathematical Association of America and Olaf Neumann, "Cyclotomy: from Euler through Vandermonde to Gauss", p.p. 323-362 in the collection "Leonhard Euler: Life, Work and Legacy" edited by Bradley and Sandifer, 2007. Stacy Langton has given a lot of details about Euler's work on the theory of equations, and also some advice on the translation; of course any mistakes are my own. If Langton ends up writing anything about Euler' and the theory of equations I would highly recommend reading it.

math.HO↗
arXiv2008

New demonstrations about the resolution of numbers into squares

Translated from the Latin original "Novae demonstrationes circa resolutionem numerorum in quadrata" (1774). E445 in the Enestrom index. See Chapter III, section XI of Weil's "Number theory: an approach through history". Also, a very clear proof of the four squares theorem based on Euler's is Theorem 370 in Hardy and Wright, "An introduction to the theory of numbers", fifth ed. It uses Theorem 87 in Hardy and Wright, but otherwise does not assume anything else from their book. I translated most of the paper and checked those details a few months ago, but only finished last few parts now. If anything isn't clear please email me.

math.HO↗
arXiv2008

Principles of the motion of fluids

This is an English translation of Euler's article "Principia motus fluidorum" in which the Euler equation (in two three dimensions) has been established for the first time in 1752. The actual publication has been delayed by nine years. Finally, this article has been published in 1761 in Novi Commentarii Academiae Scientiarum Petropolitanae.

physics.hist-ph↗
arXiv2008

Observations on a certain theorem of Fermat and on others concerning prime numbers

E26 in the Enestrom index. Translated from the Latin original, "Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus" (1732). In this paper Euler gives a counterexample to Fermat's claim that all numbers of the form 2^{2^m}+1 are primes, by showing 2^{2^5}+1=4294967297 is divisible by 641. He also considers many cases in which we are guaranteed that a number is composite, but he notes clearly that it is not possible to have a full list of circumstances under which a number is composite. He then gives a theorem and several corollaries of it, but he says that he does not have a proof, although he is sure of the truth of them. The main theorem is that a^n-b^n is always able to be divided by n+1 if n+1 is a prime number and both a and b cannot be divided by it.

math.HO↗
arXiv2008

Various considerations on hypergeometric series

E661 in the Enestrom index. This was originally published as "Variae considerationes circa series hypergeometricas" (1776). In this paper Euler is looking at the asymptotic behavior of infinite products that are similar to the Gamma function. He looks at the relations between some infinite products and integrals. He takes the logarithm of these infinite products, and expands these using the Euler-Maclaurin summation formula. In section 14, Euler seems to be rederiving some of the results he already proved in the paper. However I do not see how these derivations are different. If any readers think they understand please I would appreciate it if you could email me. I am presently examining Euler's work on analytic number theory. The two main topics I want to understand are the analytic continuation of analytic functions and the connection to divergent series, and the asymptotic behavior of the Gamma function.

math.HO↗
arXiv2008

On the sums of series of reciprocals

This translation has been published in Stephen Hawking (ed.), "God Created the Integers", published in 2007 by Running Press. There may have been some changes to the final published version and this copy. This is a translation from the Latin original, "De summis serierum reciprocarum" (1735). E41 in the Enestrom index. In this paper Euler finds an exact expression for the sum of the squares of the reciprocals of the positive integers, namely pi^2/6. He shows this by applying Newton's identities relating the roots and coefficients of polynomials to the power series of the sine function. Indeed, in other words this result is zeta(2)=pi^2/6, and Euler also works out zeta(4),zeta(6),...,zeta(12). His method will work out zeta(2n) for all n, but he does not give a general expression for zeta(2n); he gives a general expression involving the Bernoulli numbers in a latter paper.

math.HO↗
arXiv2008

A more accurate treatment of the problem of drawing the shortest line on a surface

E727 in the Enestrom index. This is a translation from the Latin original "Accuratior evolutio problematis de linea brevissima in superficie quacunque ducenda" (1779). Given a surface $pdx+qdy+rdz=0$, Euler wants to develop equations that give the geodesics on this surface. I am new to the calculus of variations, so it is not clear to me what steps follow from results that are previously known (like the Euler-Lagrange equation in the calculations) and what steps follow from earlier in this paper. I would appreciate comments from any readers who are familiar with calculus of variations.

math.HO↗
arXiv2007

On the partition of numbers into parts of a given type and number

E394 in the Enestrom index. Translated from the Latin original, "De partitione numerorum in partes tam numero quam specie datas" (1768). Euler finds a lot of recurrence formulas for the number of partitions of $N$ into $n$ parts from some set like 1 to 6 (numbers on the sides of a die). He starts the paper talking about how many ways a number $N$ can be formed by throwing $n$ dice. There do not seem to be any new results or ideas here that weren't in "Observationes analyticae variae de combinationibus", E158 and "De partitione numerorum", E191. In this paper Euler just does a lot of special cases. My impression is that Euler is trying to make his theory of partitions more approachable,. Also, maybe for his own benefit he wants to say it all again in different words, to make it clear.

math.HO↗
arXiv2007

On highly transcendental quantities which cannot be expressed by integral formulas

E565 in the Enestrom index. Translated from the Latin original, "De plurimis quantitatibus transcendentibus quas nullo modo per formulas integrales exprimere licet" (1775). Euler does not prove any results in this paper. It seems to me like he is trying to develop some general ideas about special functions. He gives some examples of numbers he claims but does not prove cannot be represented by definite integrals of algebraic functions. Euler has the idea that if we knew more about the function with the power series $\sum x^{t_n}$ where $t_n$ is the $n$th triangular number, this could lead to a proof of Fermat's theorem that every positive integer is the sum of three triangular numbers. This doesn't end of being fruitful for Euler, but in fact later Jacobi proves a lot of results like this with his theta functions. The last paragraph (§9) is not clear to me. My best reading is that there are infinitely many "levels" of transcendental numbers and that this is unexpected or remarkable.

math.HO↗
arXiv2007

Various analytic observations on combinations

E158 in the Enestrom index. Translation of the Latin original "Observationes analyticae variae de combinationibus" (1741). This paper introduces the problem of partitions, or partitio numerorum (the partition of integers). In the first part of the paper Euler looks at infinite symmetric functions. He defines three types of series: the first denoted with capital Latin letters are sums of powers, e.g. $A=a+b+c+...$, $B=a^2+b^2+c^3+...$, etc.; the second denoted with lower case Greek letters are the elementary symmetric functions; the third denoted with Germanic letters are sums of all combinations of $n$ symbols, e.g. $\mathfrak{A}=a+b+c+...$ is the series for $n=1$, $\mathfrak{B}=a^2+ab+b^2+ac+bc+c^2+...$ is the series for $n=2$, etc. Euler proves a lot of relations between these series. He defines some infinite products and proves some more relations between the products and these series. Then in §17 he looks at the particular case where $a=n,b=n^2,c=n^3$ etc. In §19 he says the Naudé has proposed studying the number of ways to break an integer into a certain number of parts. Euler proves his recurrence relations for the number of partitions into a $μ$ parts with repetition and without repetition. Finally at the end of the paper Euler states the pentagonal number theorem, but says he hasn't been able to prove it rigorously.

math.HO↗
arXiv2007

The solution of a memorable problem by a special artifice of calculation

E731 in the Enestrom index. Originally published as "Solutio problematis ob singularia calculi artificia memorabilis", Memoires de l'academie des sciences de St-Petersbourg 2 (1810), 3-9. For $z$ the distance from the origin, and $v$ a given function of $z$, Euler wants to find a curve $s$ such that the integral of $z$ over $s$ is a maximum or a minimum. He starts with the Euler-Lagrange equation, and does a lot of manipulations with polar coordinates.

math.HO↗
arXiv2007

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.

math.HO↗
arXiv2007

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 $(-1)^k$.

math.HO↗