Are there any arithmetic problems studied by Euler still open?

Question

Fermat's last theorem, which Euler had studied in the case of certain exponents, was only solved in the 1990s. Also, a counterexample to Euler's sum of powers conjecture has been found quite recently (1966). Apart from Goldbach's conjecture, are there any arithmetic problems of the like studied by Euler still open?

Answer 1

Here are some open problems that go back to Euler or, in the last case, his contemporary Goldbach.

1. The classification of all idoneal (convenient) numbers. They were introduced by Euler and he searched for them up to 10000, finding no further examples after 1848. That they seemed to stop appearing surprised Euler. It is known now that there are only finitely many idoneal numbers, and Weinberger showed GRH implies 1848 is the biggest idoneal number. As long as GRH is an open problem, the completeness of the list of idoneal numbers may remain an unsolved problem.

2. The existence of a perfect Euler brick. This is not an important problem, and as far I am aware it is not related to anything important, in contrast to the link between the old congruent number problem and the Birch and Swinnerton-Dyer conjecture.

3. The irrationality of Euler’s constant $.577...$, a number which Euler introduced in 1734 here (or in English here) and computed to more accuracy here (or in English here). Any time he studied new numbers, he tried to relate them to what was known, like trying to write them in terms of $\pi$ or logarithms. In the second paper, after computing Euler's constant to $16$ digits (with a mistake only in the last digit he found), he called this number "even more remarkable since it has still not been possible for me in any way to reduce it to a certain known measure." I think a charitable interpretation of that comment means he suspected Euler's constant is at the very least irrational, if not transcendental. Those are the kinds of things he thought about: he had proved $e$ is irrational only a few years after introducing Euler's constant and he raised the question of whether $\pi$ is transcendental in 1775, a year before presenting the second paper about computing Euler's constant.

4. Infinitude of primes of the form $a^2 + 1$. In October 1752, Euler computed all $a \leq 1500$ for which $a^2 + 1$ is prime. See pp. 587-588 here, where his list has only 3 errors: it omits $a=1080$ and mistakenly includes $a = 844$ and $1234$. Euler kept finding prime values up to $a = 1494$, and even though he does not explicitly say he conjectures there are infinitely many such primes, what he does write makes it quite clear that he thinks it is the case.

5. Writing odd numbers as $p + 2a^2$. In November 1752, Goldbach wrote to Euler with a conjecture that every odd number bigger than $1$ has the form $p + 2a^2$ for some prime $p$ and integer $a \geq 0$. See it here. (In that era $1$ was considered prime, so Goldbach's claim allowed $1$ to be so represented using $p=1$ and $a = 0$ and he'd allow $9 = 1 + 2 \cdot 2^2$ while we'd only allow $9 = 7 + 2 \cdot 1^2$.) Euler replied here that he checked this for all odd numbers below $1000$. It was discovered by Stern and his students in 1856 that there are two counterexamples: $5777$ and $5993$. (These numbers don't even fit the pattern if we allow $1$ to be a prime!) Since there are two counterexamples you could say the conjecture is simply wrong, but that's a bit harsh: wouldn't it be interesting if $5777$ and $5993$ are the only counterexamples? Indeed, Hardy and Littlewood's Conjecture I on page 49 of their paper Partitio Numerorum III here conjecture that every sufficiently large odd number $n$ has the form $p + 2a^2$ and they suggest a growing asymptotic estimate for the number of representations of $n$ in that form. An account of this problem for a general audience appeared in Mathematics Magazine here. This conjecture of Hardy and Littlewood should be regarded as the correct version of Goldbach's question, and it is still not solved.