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No, there is no such ellipse. This is exactly theorem $6.5$ of Alan Baker's book TRANSCENDENTAL NUMBER THEORY, as pointed out by Felipe Voloch. Let $\omega$ be a primitive period of a $\wp$-function …

Here is a remark, if $\chi$ is a quadratic character. Then there is a remarkable result of Gauss, saying that $$ G(\chi, N)=\begin{cases} \sqrt{N}, \;\text{ if } N\equiv 1 (4) \cr 0, \; \text{ if } N …

The main application of the (generalized) Thue-Siegel-Roth theorem to exponential Diophantine equations comes from Schmidt's subspace theorem, and an extension to $p$-adic valuations due to H.P. Schli …

There are several good inequalities for this difference, e.g., $$ n \log n + n(\log \log n -1)<P(n) < n \log n + n\log \log n $$ for all $n\ge 6$, which can be derived from the prime number theorem. B …

I just want to make a comment, that one can in general find a lot of information on the (primitive) integer solutions of the generalized Fermat equation $Ax^p+By^q=Cz^r$ (depending on the three cases …

Yes, counterexamples are easy to construct. Take $p=3$ and $n=7$. Then $3^7-1=2186=2\cdot 1093$ with the prime $1093$. Hence $\phi(3^7-1)=\phi(1093)=1092=2\cdot 3\cdot 7\cdot 13$ is divisible by $3$. …

This is not an answer, but just an observation, which Martin had probably in mind. By Fermat we know that $n=x^2+y^2$ is the sum of two squares iff the primes $p$ dividing $n$ with $p\equiv 3$ mod $4 …

I am not an expert here, but I think it has been implicitly considered in the context of $k$-Lehmer numbers. These are the positive composite integers $n\ge 1$ which satisfy $\phi(n)\mid (n-1)^k$, whe …

EDIT: I correct my answer. We have $3^n -2^n\equiv 1 \mod p$ for the smallest prime divisor $p$ of $n$. The proof is the same as the one by Gjergji Zaimi. Write $n=p^k\ell$ and use Fermat's little th …

If $G$ is a group of order $n$ with $gcd(n,\phi(n))=1$, then $G$ is cyclic. Conversely, if $n$ is an integer, such that every group of order $n$ is cyclic, then $gcd(n,\phi(n))=1$. So, counting these …

There is the following result of Wolke from $1967$ (which is perhaps not the best, but quite good). Theorem: Let $p$ be an odd prime, and $L(s,\chi)$ the $L$-series for the Dirichlet character $(n/p)$ …

There is a nice survey article, Paul Erdös and Egyptian Fractions by R.L. Graham, also referring to the well-known Erdős–Straus conjecture.

It is well known that $\sum_{k\le x} \phi(k)=\frac{3}{\pi^2}x^2+O(x\log(x))$. In a similar way we obtain $$ \sum_{k\le x} \frac{\phi(k)}{k}=\frac{6}{\pi^2}x+O(\log(x)), $$ by using the Moebius functio …

You can just recursively compute the gcd of $t\ge 2$ polynomials by $$ \gcd(P_1, P_2, \dots , P_t) = \gcd( P_1, \gcd(P_2, \dots , P_t)), $$ starting with $\gcd(P_{t-1},P_t)$, $\gcd(P_{t-2}, \gcd(P_{t- …

Euler wrote $a^4-d^4=c^4-b^4$ with $a=p+q, d=p−q, c=r+s$ and $b=r−s$, obtaining $pq(pp + qq) = rs(rr + ss)$, and then did several other special transformations, until he arrived at the special solutio …