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Leonhard Euler

Publications and source records attributed to Leonhard Euler.

At least 37 records · Page 2Linked to original sources
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.

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arXiv2010

Various observations on angles proceeding in geometric progression

This is a translation of Euler's 1773 "Variae observationes circa angulos in progressione geometrica progredientes", E561 in the Enestr{ö}m index. I translated this paper as a result of my study of Euler's work on the infinite product $\prod_{k=1}^\infty (1-z^k)$. If one instead considers the finite product $\prod_{k=1}^n (1-z^k)$, one can study its behavior on the unit circle. The absolute value of $\prod_{k=1}^n (1-e^{ikθ})$ is $2^n |\prod_{k=1}^n \sin kθ/2|$. My interest in the product $\prod_{k=1}^n \sin kθ/2$ has inspired me to become acquainted with Euler's papers on trigonometric identities, in particular E447, E561, and E562. E561 says nothing about the product $\prod_{k=1}^n \sin kθ/2$, but it has identities which I had not seen before. The identities have a form similar to Viète's infinite product $\prod_{k=1}^\infty \cos θ/2^k=\frac{\sinθ}θ$.

math.HO↗
arXiv2010

An algebraic problem of finding four numbers given the products of each of the numbers with the sum of the other three

This is a translation from the Latin of Euler's "Problema algebraicum de inveniendis quatuor numeris ex datis totidem productis uniuscuiusque horum numerorum in summas trium reliquorum", Opera Postuma 1 (1862), 282-287, reprinted in Leonhardi Euleri Opera omnia I.6. This paper is E808 in the E808 in the Eneström index. We are given the four products $a=v(x+y+z), b=x(v+y+z), c=y(v+x+z), d=z(v+x+y)$. We want to determine $v,x,y,z$ from this data. Euler wants to do this symmetrically: instead of successively eliminating variables and finding out, say, z in terms of a,b,c and then finding y, and then finding x, and then finding v, Euler introduces a new variable that has the same relation with each of the unknowns v,x,y,z and works with that. Euler finds that there will be up to eight choices of v,x,y,z for each choice of a,b,c,d.

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arXiv2009

Expansion of the infinite product $(1-x)(1-xx)(1-x^3)(1-x^4)(1-x^5)(1-x^6)$ etc. into a simple series

Translated from the Latin original "Evolutio producti infiniti $(1-x)(1-xx)(1-x^3)(1x^4)(1-x^5)(1-x^6)$ etc. in seriem simplicem" (1775). E541 in the Enestroem index. In this paper Euler is revisiting his proof of the pentagonal number theorem. He gives his original proof explained a bit differently, and then gives a different proof. However this second proof is still rather close to his original proof. To understand the two proofs, I wrote them out using subscript notation and sum/product notation. It would be a useful exercise to try to really understand the proofs without using any modern notation. The right notation takes care of a lot for us, which we would otherwise have to keep active in our minds.

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arXiv2009

On amicable numbers

Translated from the Latin original, "De numeris amicabilibus" (1747). E100 in the Enestroem index. Euler starts by saying that with the success of mathematical analysis, number theory has been neglected. He argues that number theory is still a worthwhile subject. First, because there are very elegant and difficult questions in it. Second, because some of the greatest figures in mathematics so far had spent time working on problems in number theory. In particular, Descartes spent time looking for (and finding) a pair of amicable numbers that previously in Europe had been unknown. Euler also mentions the work of Frans van Schooten. Then Euler explains the form of the amicable numbers that were then known, which were $2^n xy$ and $2^n z$, where $x,y,z$ are prime, and the two conditions i) $z=xy+x+y$ and ii) $2^n(x+y+2)=xy+x+y+1$ are satisfied. Euler says that to find more amicable numbers different forms should be tried. He gives 30 pairs of amicable numbers but does not explain how he found them. This is discussed in Ed Sandifer's November 2005 "How Euler did it". One of the pairs (pair XIII) Euler lists is incorrect.

math.HO↗
arXiv2009

Observations on two fourth powers whose sum is equal to the sum of two other fourth powers

Translated from the Latin original, "Observationes circa bina biquadrata quorum summam in duo alia biquadrata resolvere liceat" (1772). E428 in the Enestroem index. This paper is about finding A,B,C,D such that $A^4+B^4=C^4+D^4$. In sect. 1, Euler states his "quartic conjecture" that there do not exist any nontrivial integer solutions to $A^4+B^4+C^4=D^4$. I do not know whether he stated this conjecture previously. In sect. 3, Euler sets A=p+q, B=p-q, C=r+s,D=r-s. Taking r=p and s=q gives the trivial solution of C=A and B=D, but this gives Euler the idea of making p and r multiples of each other and q and s multiples of each other. If k=ab this again gives the obvious solution, so in sect. 5: Perturb k to be $k=ab(1+z)$. Euler works out two solutions. One is A=2219449, B=-555617, C=1584749, D=2061283. Hardy and Wright, fifth ed., p. 201 give a simpler parametric solution of $A^4+B^4=C^4+D^4$. Thomas Heath in his Diophantus, second ed., pp. 377-380 gives a faithful explanation of Euler's solution.

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arXiv2009

An arithmetic theorem and its demonstration

Translated from the Latin original, "Theorema arithmeticum eiusque demonstratio", Commentationes arithmeticae collectae 2 (1849), 588-592. E794 in the Enestroem index. For m distinct numbers a,b,c,d,...,\upsilon,x this paper evaluates \[ \frac{a^n}{(a-b)(a-c)(a-d)...(a-x)}+\frac{b^n}{(b-a)(b-c)(b-d)...(b-x)} +...+\frac{x^n}{(x-a)(x-b)(x-c)...(x-\upsilon)}. \] When $n \leq m-2$, the sum is 0, which Euler had already shown in sect. 1169 of his Institutiones calculi integralis, vol. I, E366. When $n=m-1$ the sum is $=1$. When $n \geq m$, then using Newton's identities Euler gets expressions in terms of sums of powers of the numbers a,b,c,d,...,\upsilon,x. The notation used in this paper is confus(ing/ed), and indeed the editors of the Opera omnia made several corrections to the original publication. Near the end of the paper, "... for each of them one searches for what the character will be..." is the only reading of the sentence I can come up with. Examples of a work that refers to this theorem is J.-P. Serre, Corps locaux, 2nd ed., ch. III, sect. 6 (for $n \leq m-1$), and D. E. Knuth, The Art of Computer Programming, vol. 1, sect. 1.2.3, exercise 33.

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arXiv2009

A demonstration of a theorem on the order observed in the sums of divisors

Translation from the Latin of Euler's "Demonstratio theorematis circa ordinem in summis divisorum observatum" (1760). E244 in the Enestroem index. In his previous paper E243, Euler stated the pentagonal number theorem and assuming it proved a recurrence relation for the sum of divisors function. In E244 Euler proves the pentagonal number theorem, and then repeats the proof of the recurrence. In Proposition 1, Euler proves that $\prod_{n=1}^\infty (1-a_n)=1-\sum_{n=1}^\infty a_n(1-a_1)...(1-a_{n-1})$. In Proposition 2 he explicitly states the case $a_n=x^n$. Then in Proposition 3, Euler applies Proposition 2 then collects terms: each time this leaves two terms from the series $\sum_{n=-\infty}^\infty (-1)^n x^{n(3n-1)/2}$. This is explained in detail in a paper I am working on about Euler's work on the pentagonal number theorem.

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arXiv2009

An observation on the sums of divisors

Translation from the Latin of Euler's "Observatio de summis divisorum" (1752). E243 in the Enestroem index. The pentagonal number theorem is that $\prod_{n=1}^\infty (1-x^n)=\sum_{n=-\infty}^\infty (-1)^n x^{n(3n-1)/2}$. This paper assumes the pentagonal number theorem and uses it to prove a recurrence relation for the sum of divisors function. The term "pentagonal numbers" comes from polygonal numbers. Euler takes the logarithmic derivative of both sides. Then after multiplying both sides by $-x$, the left side is equal to $\sum_{n=1}^\infty σ(n) x^n$, where $σ(n)$ is the sum of the divisors of $n$, e.g. $σ(6)=12$. This then leads to the recurrence relation for $σ(n)$. I have been studying in detail all of Euler's work on the pentagonal number theorem, and more generally infinite products. I would be particularly interested to see if anyone else worked with products and series like these between Euler and Jacobi, and I would enjoy hearing from anyone who knows something about this.

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arXiv2009

On the infinity of infinities of orders of the infinitely large and infinitely small

Translation (by J.B.) from the original Latin of Euler's "De infinities infinitis gradibus tam infinite magnorum quam infinite parvorum" (1780). E507 in the Eneström index. Euler discusses orders of infinity in this paper. In other words this paper is about how different functions approach infinity or 0 at different rates. I was not certain about what Euler means by "infinities infiniti". Probably he means that $x,x^2,x^3$, etc. are infinitely many orders of infinity, and $\log x,(\log x)^2, (\log x)^3$, etc. are infinitely many orders of infinity, so the combinations of them are an infinity of infinities of orders of infinity. In fact Euler mentions other orders of infinity in this paper. It would be worthwhile to study this paper more to figure out exactly what Euler means here. Another translation of the title is "On the infinitely infinite orders of the infinitely large and infinitely small". Here's another place Euler uses the phrase "infinities infiniti". The phrase "infinities infiniti" from the title is used by Euler also in section 21 of E302, "De motu vibratio tympanorum". Truesdell translates this phrase on p. 333 of "The rational mechanics of flexible or elastic bodies" as "infinity of infinities". I'd like to thank Martin Mattmueller for clearing up some questions.

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arXiv2009

Annotations to a certain passage of Descartes for finding the quadrature of the circle

Translation from the Latin of "Annotationes in locum quendam Cartesii ad circuli quadraturam spectantem" (1763). The passage Euler is referring to is the "Excerpta" in part 6, p. 6 of Descartes' 1701 "Opuscula posthuma". Before reading this paper I had not heard of the "quadratrix" before, and I recommend learning a bit about it before reading this. I found Thomas Heath, "A history of Greek mathematics", vol. I, chapter VII to be helpful, in particular pp. 226-230. The quadratrix is a "mechanical curve" that can be used to rectify the circle. The usual problem of squaring the circle is to construct a square with the same area (or perimeter) as a given circle, in a finite number of steps using compass and straightedge. Descartes worked in the reverse direction: from a given square he constructed the radius of a circle with the same perimeter, but in an infinite number of steps. In this paper Euler reconstructs Descartes' argument and develops some consequences of it. Euler finds that \[ \sum_{n=0}^\infty \frac{1}{2^n} \tan \frac{1}{2^n}ϕ= \frac{1}ϕ - 2\cot 2ϕ. \] Integrating this yields \[ \prod_{n=1}^\infty \sec \frac{1}{2^n} ϕ= \frac{2ϕ}{\sin 2ϕ}. \] I'd like to thank Davide Crippa from the University of Paris 7 for some helpful back and forth about this paper. One of the only citations to this paper that I have found is in Pietro Ferroni, De calculo integralium exercitatio mathematica, Allegrini, Florence, 1792, pp. xxi--xxiii. The full text of it is available on Google Books.

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arXiv2008

An inquiry into whether or not 1000009 is a prime number

Translation from the Latin original "Utrum hic numerus 1000009 sit primus necne inquiritur" (1778). E699 in the Enestrom index. The idea of this paper is that if some number is a sum of two squares in two ways, then some other smaller number must be a square. We then check all the cases. If we find a case where this smaller number is a square then we can use this to find a factor of the original number, which is therefore composite. If we check all the cases and none of them are squares then the original number can be written as a sum of two squares in only one way, and hence is prime. Euler first examines the number 1000009 and then 1000081. Obviously 1000009=1000^2+3^2, and then he finds that 1000009=972^2+235^2, hence it is not prime. On the other hand, Euler finds that 1000081 can be written as a sum of squares in only one way, hence it is prime.

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