Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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Prime spiral with regular triangles

I wrote a little program that creates a Ulam like spiral, only with regular triangles instead of squares. The following image shows the way it is set up: In this set up all the uneven numbers will ...
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On the fourth power $2^4 + 15045^4 + 26870^4 + 34090^4 = 37239^4$

The Diophantine equation, $$x_1^4+x_2^4+x_3^4+x_4^4 = z^4\tag1$$ can be solved with $x_1 = 0$ (Elkies), or as the title shows, with $x_1 = 2$ (Wroblewski). See link here. Q: Can $(1)$ be solved in ...
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Number Theory: $x^2\equiv 1\pmod{140}$

I have this problem assigned for homework and I'm confused as to how to solve an $x^2$ congruence. Here is the problem: $x^2\equiv 1\pmod{140}$ My only thought was to do something along the lines ...
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57 views

Inverse limit of $\Bbb Q/q\Bbb Z$ isomorphic to finite adeles?

Let $\Bbb Q/q\Bbb Z$, for some positive rational $q$, denote the quotient group of the discrete rationals by the subgroup of integers times $q$. For any $q_1, q_2 \in \Bbb Q^+$ and $n \in \Bbb N^+$ ...
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168 views

How big are the factors of $2^kp - 1$?

Let $p$ be a prime number, $p > 3$. Does it always exists a $k \in \mathbb N, k \ge 1$ such that the prime factors of $2^kp - 1$ are all less then $p$? Thoughts Well we can easily see that ...
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136 views

Integer part of $e^x$

Using wolfram alpha gives that $\lfloor e^{2015}\rfloor \equiv 1 \pmod 4$. Is it possible to prove this by manual ? Thank you.
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112 views

Approximating $\pi$ by an expression of the form $\sqrt{\sqrt{ \cdots \sqrt{ n!! \cdots !}}}$

Here is a problem that appeared as a prize challenge in a periodical for science students, back when I was a student: Find an approximation of $\pi$ formed of the numbers $0$ through $9$, each used ...
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118 views

Neukirch's motivation for $p$-adic numbers

I've started reading Neukirch's Algebraic Number Theory book and at the beginning of Chapter II he starts his motivation for the $p$-adic numbers as follows: "The idea originated from the observation ...
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108 views

How prove that for a given integer $k$ there exist infinitely many $n$ such that $S(2^n+n)+k<S(2^n)$?

Let $S(n)$ be the sum of digits of $n$. How prove that for a given integer $k$ there exist infinitely many $n$ such that $S(2^n+n)+k<S(2^n)$? I have modified this problem: Denote by $S(a)$ the ...
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160 views

Is there a quicker way to generate integers which are hard to factor than multiplying two large primes?

An easy way to generate an integer which is hard to factor is to find two large primes and multiply them. As a bonus, you know the factors. I'm interested in whether it's possible to find integers ...
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59 views

A sequence avoiding 3-term power progressions

Rankin1 studied sequences of integers that avoid 3-term geometric progressions, $(a, a c, a c^2)$, e.g., $$\{1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, \ldots \} \;$$ So, $18$ is excluded ...
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181 views

Understanding an example of a prime-like non-finite additive basis

In this answer on MO, the user Gene S. Kopp gives an example of a relatively "big" set $A\subset \mathbb{N}$ with relatively "small" gaps that fails to be an asymptotic finite basis. I'm having a ...
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48 views

Why are there palindromic subsequences at random among this sequence?

So I was thinking about the Goldbach conjecture and I rephrased it to myself as the following: Prove that every number lies halfway between two primes (or is itself prime.) Which is equivalent. ...
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175 views

Summation of a function 2

Let $n$ is a positive integer. $n = p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is the complete prime factorization of $n$. Let me define a function $f(n)$ $f(n) = p_1^{c_1}p_2^{c_2}\cdots p_k^{c_k}$ where ...
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109 views

The smallest non-zero integer $c$ such that $\sum\limits_{n=1}^m\pm(x+n)^6 = c$?

We have the neat equalities, I. Group 1 For $k=2,3,4,5,\dots$ $$\sum_{n=1}^{2^k}\epsilon_n(x+n)^k = 2^{\frac{k(k-1)}{2}}k! = 4,\;48,\;1536,\;\color{brown}{122880},\dots$$ for appropriate ...
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Official name of Fermat's $x^2+3y^2$ theorem?

One of Fermat's more well-known claims is that for any prime number $p$, $p=x^2+3y^2\iff p\equiv 1\pmod 3$. Does this have an "official" name? (Another one which goes $p=x^2+y^2\iff p\equiv 1\pmod 4$ ...
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Is there any elegant formalization of fractional numbers?

The question is just what is on the title, but I'll describe the context for completion: Natural numbers can be encoded quite elegantly on the Lambda Calculus as church numbers, that is, a function ...
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Kloosterman sum and multiples of 16

A Kloosterman sum is defined as $$K(a,b;m)=\sum_{0\leq x \leq m-1}_{\gcd(x,m)=1} e^{2\pi \mathcal{i} (ax+bx^*)/m}$$ where $a,b,m \in \mathbb{N}$ and $x^*$ is the inverse of $x$ modulo $m$. How can ...
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Numbers Made From Concatenating Prime Factorizations

I came across the following curious problem while playing around with my calculator. Take any positive integer $n$; for this example we'll use $216$. Create a sequence as follows: Factor $n$ into ...
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223 views

Does this prime generating way generate all the prime numbers?

I've thought of the following algorithm to find the entire list of prime numbers: Take a prime number $p$ to your list. $1.$ Multiply all the numbers in your list and call the number you get ...
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54 views

When does a variety have a point over a finite field for sufficiently large primes p?

Let $X$ be an algebraic variety over the rational numbers. Suppose that $X$ has positive dimension. I would like to say that $X(\mathbb{F}_p)$ is non-empty for sufficiently large primes $p$. One idea ...
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Diophantine: $x^3+5=y^5$

Find all integers $x$ and $y$ such that $x^3+5=y^5$. I found this after solving the equation $3^a+5=2^b$. For this equation, since $(a,b)=(3,5)$ is a solution, it is possible to write it as ...
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Binomial triplets

Solutions to the equation $$\dbinom{a}{n}+\dbinom{b}{n}=\dbinom{c}{n}$$ I will refer to as 'Binomial triplets of order $n$'. These triplets describe simplicial $n$-polytopic numbers that can be ...
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46 views

Do modular forms of higher weight occur in the proof of FLT?

Fermat's last theorem is a consequence of a statement about weight two modular forms. Going through the long proof, does one ever encounter modular forms of higher weight?
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264 views

Successive ratios of a sequence, is this limit true?

The natural numbers $1,2,3,4,5....$ can be calculated as the row sums of the triangle $T(n,k)$ equal to $1$ if $n \geq k$ and $0$ otherwise: $$\displaystyle T = \left(\begin{matrix} ...
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finding very small values of $r + s\sqrt p + t\sqrt q$

For integers $p$, $q$ I'm interested in finding good approximations to $0$ of the form $r + s\sqrt p + t\sqrt q$ for rational $r$, $s$ and $t$. In particular, I'd like to generate approximations ...
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English translation of two papers by Polya on real zeros of Fourier transform approximation to Riemann $\xi$ function

I am looking for English translation of the following two papers by Polya: [1] G. Polya, Bemerkung über die Integraldarstellung der Riemannschen $\xi$-Funktion, Acta Math. 48(1926), 305-317; ...
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How find this $aA_{m+1}=\overline{\sigma_{0}\sigma_{1}\sigma_{2}\cdots\sigma_{m}}$

Question let $m$ is positive numbers,and such $m\ge 5$,and $$A_{m+1}=\overline{1234\cdots m}=1\times (m+1)^{m-1}+2\times (m+1)^{m-2}+\cdots+(m-1)\times (m+1)+m$$(or see ...
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104 views

Number of points on an affine hypersurface

I am curious to know the following and I would appreciate any help! Let $k$ be a finite field of $q$ elements. Let $X \subseteq \mathbb{A}^{n+1}_k $ be the affine hypersurface defined by $f = ...
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Prime number theorem for Dedekind domains

Let $\mathscr P\subseteq \mathbb N$ be the set of prime numbers. The prime number theorem tells us that if $\pi(x)=|\{p\in\mathscr P\colon p\leq x\}|$ then $\pi(x)\sim \frac{x}{\log x}$. Now one could ...
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Are $ut + 1$ and $ut + t + 1$ both prime for some t for any $u$?

Conjecture : For any natural number $u$, there is a natural number $t$ such that $ut + 1$ and $ut + t + 1$ are both prime. So we get a solution of the equation $$au - b(u+1) = -1$$ with prime ...
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131 views

Role of the discrete subgroups of Lie groups

This is a question I don't believe is too vague to admit a sensible answer: In what directions can the structure theory of the discrete subgroups of real and p-adic Lie groups applied? What ...
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161 views

Ramification index and residue class degree under completion

I've got a problem in proving something written at page 111 of the book "Algebraic Number Theory" by A. Fröhlich and M. J. Taylor. This is the setting. Let $\mathfrak{o}$ be a Dedekind domain with ...
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384 views

Can $\sigma_4(n)$ be a square number?

Denote $\sigma_k(n)=\sum_{d\mid n}d^k,$ then $$\sigma_1(3)=2^2,\sigma_2(42)=50^2,\sigma_3(2)=3^2,$$ and both $\sigma_1(345)$ and $\sigma_3(345)$ are square numbers: ...
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103 views

Pairwise sums are equal

The distinct positive integers $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ with $n\ge2$ have the property that the $\binom{n}2$ sums $a_i+a_j$ are the same as the $\binom{n}2$ sums $b_i+b_j$ (in some order). ...
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331 views

Which unordered partition of $n$ gives rise to the largest number of ordered partitions?

A quick look at the wikipedia article on partitions of $n \in \mathbb{N}$ shows that the number of ordered partitions is $2^{n-1}$, and the number of unordered partitions is asymptotically $ \sim ...
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181 views

How to partition $S$ in this way?

Assume: $$ P =\{p_1,p_2,\cdots,p_K\}\subset \{1,2,\dots,N\},\quad |P| = K, \qquad x \in \mathbb{R}_+^K , \qquad w = e^{-j\frac{2\pi}N} $$ and, $$ f(l) = \sum_{i=1}^K \sum_{j=1}^K x_i x_j ...
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Constructing pairs of units $(x,y)$ which solve $x^2 + y^2 \equiv -1 \pmod{N}$

A classic result on the way to the Lagrange Four Squares theorem — for instance proven by Theorem 87 of Hardy & Wright, as noted by this remark on the Four Squares theorem — is that ...
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Relative density of images of diophantine polynomials

My current research project involves sampling a sequence along non-constant polynomials with non-negative integer coefficients defined over the natural numbers, and I'd like to be able to say when two ...
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253 views

How Ramanujan find this formula

I have seen this formula from Ramanujan $\sum_n \frac{a^{n+1}-b^{n+1}}{a-b}\frac{c^{n+1}-d^{n+1}}{c-d}T^n=\frac{1-abcdT^2}{(1-abT)(1-acT)(1-bcT)(1-bdT)}$. I know how to prove it via geometric ...
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Matrix representation of field automorphism

Let $K$ be the degree $n$ field extension of the field $k$, and let $\alpha_1,\dots,\alpha_n\in K$ be a basis of $K$ over $k$ (as a vector space). I read somewhere that the following matrix $$ M= ...
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Understanding Newman's proof of the prime number theorem

I am trying to understand D.J. Newman's proof of the prime number theorem, as presented by D. Zagier. I am not too familiar with analysis, and so there are some things I don't understand. In (III), ...
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356 views

Maximum length of sequence of non-coprimes of $N$ - least upper bound for Jacobsthal's function

I am looking at the length of the longest sequences of adjacent integers that are not coprime to $N$ for very large $N$. Let $F_N$ be the set of integers less than $N$ which are not coprime with $N$: ...
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130 views

Inverting the Riemann zeta function in $s>1$

Let $s>1$ be a positive real and the Riemann zeta fucntion be defined for $s>1$ as $$ \zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^s}. $$ I am looking for an inversion formula for the zeta ...
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Best upper bound on the number of divisors of $n$ that are larger than $N$.

I am looking for the best upper bound on $$\sum_{\substack{d | n\\ d \geq N}} 1.$$ I know that $$ d(n) = \sum_{\substack{d | n}} 1 \leq e^{O(\frac{\log n}{\log \log n})}. $$ For my application, I ...
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158 views

Each new member from the second divides the sum of all previous

All integers from $1$ to $13$ are recorded in a sequence such that each number (from the second onwards) divides the sum of all previous numbers. What numbers can be in the third place and why? I ...
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393 views

The divergence of the series of reciprocals of primes (proof check):

I just wanted to check my attempt at a proof for the divergence of: $$\sum_{n=1}^{\infty} \frac{1}{p_n} \tag{ $\star$ }$$ We begin with assuming that $(\star)$ converges. If $(\star)$ ...
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130 views

A formula by R.L.Graham,AMM(1995)

I see this formula in a book, it comes from R.L.Graham,AMM(1995) : ...
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144 views

Proof that $t^8+2t^6+4t^4+t^2+1$ is reducible in $\mathbb{F}_p$

Prove that the polynomial $t^8+2t^6+4t^4+t^2+1$ is reducible in $\Bbb F_p$, for all $p\in \Bbb P$. Here are some examples: $t^8+2t^6+4t^4+t^2+1=(1 + t + t^4)^2\pmod{2}$ $t^8+2t^6+4t^4+t^2+1=(1 + t) (2 ...
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Primes of the form $\frac{n^2-n+4}{2}$ satisfy Hardy-Littlewood analogue?

Let $n,a,b$ be positive integers with $a<b$. Consider primes of the form $f(n)=\dfrac{n^2-n+4}{2}$. Let $C(a,b)$ denote the amount of primes of the form $f(n)$ between (and including) $f(a)$ and ...