# Questions tagged [nt.number-theory]

Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions

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### Two conjectures about generalised A329369

Let $m \geqslant 2$ be a fixed integer. Let $$\operatorname{wt}(n,m)=\operatorname{wt}\left(\left\lfloor\frac{n}{m}\right\rfloor,m\right)+n\bmod m, \operatorname{wt}(0,m)=0$$ Then we have an integer ...
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### Ambiguity about the exact definition of coefficients of modular forms

You can see the parts after my questions in the boxes. I received the answer to my first question in the comments. I am confused about the definition of $a_n$ and $b_n$ in Part II below. I know the ...
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### Primes that are sums of two squares with constraints on the squares

It is well known that there are infinitely many primes of the form $a^2+b^2$ (namely all primes congruent to $1$ modulo $4$). On the other hand, Euler raised the problem as to whether there are ...
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### Can a lower bound for this weakening of Goldbach's conjecture be reached?

Say a non negative integer $r$ is a primality radius of $n$ if both $n-r$ and $n+r$ are prime, and that a non negative integer $w$ is a weak primality radius of $m$ if $\omega(m-w)=\omega(m+w)=1$, ...
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### Modulo $2$ binomial transform of A243499 applied $k$ times

Let $m \geqslant 1$ be a fixed integer. Let $f(n)$ be A007814, exponent of highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$. ...
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### Are Li's numbers $\lambda_n$ absolutely convergent for $n>1$?

Li's numbers $\{\lambda_n\}$ are defined as $$\lambda_n=\frac{1}{(n-1)!}\frac{d^n}{ds^n} [s^{n-1}\log\xi(s)]_{s=1}$$ for all positive integers $n$. Also $\lambda_n$ is given as a sum over the non ...
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### Magic hourglass of squares hyperelliptic equation

I have been looking into the problem of the magic square of squares, or more specifically, the magic hourglass of squares, like so: $a^2$ $b^2$ $c^2$      $d^2$ $e^2$ $f^2$ $g^2$ ...
### Modulo $2$ binomial transform of A124758
Let $f(n)$ be A153733, remove all trailing ones in binary representation of $n$. Here \begin{align} f(2n)& = 2n\\ f(2n+1)& = f(n)\\ \end{align} Then we have an integer sequence given by \begin{...
A factorial prime is of the form $n! \pm 1$. The first $14$ factorial primes are listed in the Online Integer Sequences (OEIS) A088054:  2, 3, 5, 7, 23, 719, 5039, 39916801, 479001599, 87178291199, ...