From The Editors
1 March 2026
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Volume 6, Issue 1 (2026)Read More
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- Articles & Notes1 March 2026
How Euler Could Have Done It: Euler and a Heuristic Derivation of the Prime Number Theorem
We present an argument that leads to the formulation of the Prime Number Theorem that Euler could have given.
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- From The Editors1 March 2026
Historical Beginnings to Modern Results
An introduction to the contents in Volume 6 (Issue 1) of Euleriana. - Translation & Commentary1 March 2026
About Infinite Algebraic Curves whose Indefinite Lengths Equal an Elliptic Arc: An English Translation of E780
Euler continues his study of algebraic curves with equal arc length, a subject to which he returned several times. After a brief review, he introduces an infinite family of curves with the same indefinite length as a given ellipse. However, this first family is by his own admission poorly motivated, so he derives directly a different but related family of curves with the same indefinite length as a given ellipse. - Translation & Commentary1 March 2026
The Extent To Which the Earth's Motion is Perturbed by the Moon, More Accurately Investigated: An English Translation of E139
In Tabulae astronomicae solis et lunae (Solar and lunar astronomical tables) (E87), published in 1746, Euler made a first attempt to correct his astronomical tables for the gravitational attraction of the Earth toward the Moon, simply by accounting for the difference between the position of the Earth and the center of gravity of the Earth-Moon system. In this article Euler revisited the problem, noting that this center of gravity would not itself take an elliptical path around the Sun. To model the motion more accurately, Euler considered the separate gravitational attractions of the Sun and Moon acting on the Earth. With this approach, Euler found expressions dependent on the angle between the Sun and Moon as seen from Earth that can be used to correct the direction of the Sun as seen from Earth as well as the distance from the Earth to the Sun. Unfortunately for Euler, the accuracy of these corrections depends on accurate estimates of the density of the Moon and the mean distance from the Earth to the Sun, neither of which were available. - Articles & Notes1 March 2026
Euler and Gauss on the Hypergeometric Function -- A Comparison
We present several results stated by Euler on what is today called the Gaussian hypergeometric function. For this purpose, we compare the results found by Gauss to Euler's results. - Articles & Notes1 March 2026
Even Repdigits are not Perfect: Conclusions from Triangular Numbers
Are there nontrivial repdigits that are perfect numbers? This note gives an answer: not for numbers smaller than $10^{1500}$. This finding follows from results for triangular numbers. Passing by, I review Euler's contributions on polygonal numbers. - Articles & Notes1 March 2026
The Modern History of the Basel Problem
The \emph{Basel problem} refers to the problem of determining a closed form for the infinite series $\frac{1}{1^2} + \frac{1}{2^2} + \cdots$. If we consider what mathematical results have the most peer-reviewed papers devoted to new ways of proving such results, Euler's formula $\frac{1}{1^2} + \frac{1}{2^2} + \cdots = \frac{\pi^2}{6}$ is certainly among the top of such results. This motivates our historical study of peer-reviewed papers based on proofs of Euler's formula, and we introduce what appears to be the most comprehensive and up-to-date and exhaustive catalogue of peer-reviewed, published papers in the 20th and 21st centuries devoted to or mainly concerning proofs of $\frac{1}{1^2} + \frac{1}{2^2} + \cdots = \frac{\pi^2}{6}$. We show how the modern history of the Basel problem relates to many of Euler's original publications, including E41, E61, E63, E72, E421, and E592, and we categorize modern solutions to the Basel problem according to the techniques involved, and we apply our findings as a basis for our historical arguments and analyses.