Suppose $X$ is a smooth family of algebraic varieties over the base $B:=\mathbb{P}^1\backslash\lbrace0,1,\infty\rbrace$ over $\overline{\mathbb{Q}}$; then we can form the relative …

Let φ be the golden ratio, (1+√5)/2. Taking the fractional parts of its integer multiples, we obtain a sequence of values in (0,1) which are in some sense "evenly distrib …

In a comment on Tom Goodwillie's question about relating the Alexander polynomial and the Iwasawa polynomial, Minhyong Kim makes the cryptic but tantalizing statement: In brief, t …

I've been reading the paper "LARGE SPACES BETWEEN THE ZEROS OF THE RIEMANN ZETA-FUNCTION" by S. H. Saker (arXiv-0906.5458v3 [math.NT]). http://arxiv.org/abs/0906.5458. The author e …

It can be seen here that the only numbers for which $n^{m+1}\equiv n \pmod{m}$ is true are 1, 2, 6, 42, and 1806. Through experimentation, it has been found that $\displaystyle\sum …

There are several questions in the Euler-Goldbach correspondence that I am unable to answer. Sometimes it does not take very much: in his letter to Goldbach dated June 9th, 1750, …

We say that a linear form $f=ax+by+cz$ of real coefficients is "irrational" if $a,b,c$ are linearly independent over the rationals. My question is: Are there 3 such "irrational" li …

Suppose there is a function f(x) which is the "probability" that the integer x is prime. The integer x is prime with probability f(x), and then divides the larger integers with pro …

Note: I tried asking this on math.stackexchange (here), but didn't really receive an answer - so I figured this might be the right place. How can I find the number of k-permutat …

There is a well-known theorem of Shafarevich that given a finite set $S$ of primes the number of isomorphism classes of elliptic curves over $\Bbb Q$ with everywhere good reduction …

The famous Beilinson's conjecture predicts a relationship between the regulator map in $K$-theory and special value of $L$-function generalizing the Dirichlet's theorem in number t …

This may be subjective, but does anyone have any insight into why this is the case? This struck me while considering that it's also the eigth Mersenne prime (2^31-1=2147483647). …

I am curious if somebody can be helpful concerning the following experimental observation: There exist two rational sequences $\alpha_0,\alpha_1,\dots$ and $\beta_0,\beta_1,\dots …

I recall that the Manin constant for a strong elliptic curve is a rational integer $c_E$ such that, for a modular parametrization $\phi: X_1(N) \to E$, one has $\phi^*(\omega_E)= 2 …

There are many fast algorithms (deterministic and probabilistic) for detecting primality. Are there any fast algorithms (probabilistic ones allowed) known for detecting whether a n …