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Full-Text Articles in Physical Sciences and Mathematics
Euler Archive Spotlight: Multiple Search Options, Christopher Goff
Euler Archive Spotlight: Multiple Search Options, Christopher Goff
Euleriana
The Euler Archive houses PDF versions of almost all of Euler's original publications. While most visitors search the archive via a work's Eneström number, the Archive can be searched via source publication name, date written, or decade of publication. The Archive also provides context for Euler's publications through short pieces of historical information.
Euler And A Proof Of The Functional Equation For The Riemann Zeta-Function He Could Have Given, Alexander Aycock
Euler And A Proof Of The Functional Equation For The Riemann Zeta-Function He Could Have Given, Alexander Aycock
Euleriana
We explain how Euler could have proved a functional equation, which is equivalent to the one for the Riemann zeta-function, that he conjectured in his paper {\it ``Remarques sur un beau rapport entre les series des puissances tant directes que reciproques"} \cite{E352} (E352: ``Remarks on the beautiful relation between the series of the direct and reciprocal powers").
Euler And The Gaussian Summation Formula For The Hypergeometric Series, Alexander Aycock
Euler And The Gaussian Summation Formula For The Hypergeometric Series, Alexander Aycock
Euleriana
We show that in his paper {\it ``Plenior expositio serierum illarum memorabilium, quae ex unciis potestatum binomii formantur"} \cite{E663} (E663: ``A more thorough exposition of those memorable series that are formed from the binomial coefficients") Euler could have found the Gaussian summation formula for the hypergeometric series from his own formulas in that same paper, if he actually set the task for himself.
Euler And Homogeneous Difference Equations With Linear Coefficients, Alexander Aycock
Euler And Homogeneous Difference Equations With Linear Coefficients, Alexander Aycock
Euleriana
We present a method outlined by Euler in his paper{\it ``De fractionibus continuis observationes"} \cite{E123} (E123: ``Observations on continued fractions") that can be used to solve homogeneous difference equations with linear coefficients. We will illustrate his ideas by applying it to two familiar examples and explain how it can be understood from a more modern point of view.
On The Cases In Which The Formula X^4+Kxxyy+Y^4 Can Be Reduced To A Square, Georg Ehlers
On The Cases In Which The Formula X^4+Kxxyy+Y^4 Can Be Reduced To A Square, Georg Ehlers
Euleriana
Euler’s key idea for equating the Quartic in the title to a square is to set k=P+surd(Q). From this he derives P=f·x^2 and Q=4f·y^2+4 and solves the Pell equation for y. He then discusses various extensions to rational numbers that leave k an integer. Euler provides incomplete tables for integers k with |k|square.
Research On Arithmetic, Erik R. Tou
Research On Arithmetic, Erik R. Tou
Euleriana
In this English translation, some of Joseph-Louis Lagrange's early number theory is presented. Here, he laid out a theory of binary quadratic forms with special attention to the representation problem: determining those integers which may be represented by a given form, and cataloguing the possible forms of their divisors.
Number Theory And More, Christopher Goff, Erik Tou
Number Theory And More, Christopher Goff, Erik Tou
Euleriana
An introduction to the contents in Issue 1, Volume 4 of Euleriana.