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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7 views

Probability number is divisible by half the square of a prime?

Let $p$ be a prime. What is the probability that a number of the form $\left \lceil \frac{p^2}{2} \right \rceil$ divides a random positive integer $n$. I have a solution that involves the Riemann-Zeta ...
3
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0answers
16 views

Does there exist a prime number $p$ such that $p\cdot \mathcal{O}_{K}$ in $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$ is a prime ideal?

Problem: Prove or disprove: there exists a prime number $p$ such that $p \cdot \mathcal{O}_{K}$ in $K=\mathbb{Q}(\sqrt{2},\sqrt{3})$ is a prime ideal, where $\mathcal{O}_K$ denotes the ring of ...
1
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1answer
29 views

Finding reminder of negative number

Recently my colleague ask one mathematical question which is, What is the quotient and remainder of $(-29)/7$? and my answer was that quotient is $4$ and remainder is $-1$ and he told me I'am ...
-1
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0answers
30 views

How could one invert this summation?

In this paper, under Stirling Numbers and their Asymptotics, the author takes equation (3.1): ...
1
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0answers
26 views

Inequality of logarithm of the tail of the Euler product

I would want to know if this inequality holds: Let $x$ be a positive integer and $b>1$ be a real, then $$ \sum_{p> x}\log(1-p^{-b})\ge-\frac{b}{x}, $$ where the sum is over all prime $p\ge x$. ...
1
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2answers
26 views

Variation of Division Algorithm

How to derive this version of the division algorithm . For integers a, b with b ≠ 0 there exist unique integers q and r that satisfy a = qb + r, where -1/2|b| < r ≤ 1/2 b. I started off with ...
2
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0answers
30 views

Representations of $\mathbb{H}^{\times}$ and $\mathbb{H}^{\times}/\mathbb{R}^{\times}$.

In an attempt to recapture Eichler's theta correspondence I have hit a stumbling block. Let $D$ be a quaternion algebra over $\mathbb{Q}$, ramified at $p,\infty$. Also let $V_j = ...
-8
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1answer
74 views

I have written proof of Goldbach's conjecture How do I register it? [on hold]

I have written proof of Goldbach's conjecture How do I register it? Please email the appropriate steps and magazines. Thanks
3
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3answers
73 views

Show $\frac{(2n)!}{n!\cdot 2^n}$ is an integer for $n$ greater than or equal to $0$

Show $$\frac{(2n)!}{n!\cdot 2^n}$$ is an integer for $n$ greater than or equal to $0$. Could anyone please help me with this proving? Thanks!
0
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0answers
36 views

Sum of the reciprocals topology

Let's define this topology in $\Bbb N$ (here $\Bbb N$ begins at $1$): $$K\subset\Bbb N\text{ is closed }\iff K=\Bbb N\;\text{ or }\;\sum_{n\in K}\frac1n\text{ converges}$$ I have worked some on it. ...
-7
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2answers
54 views

Very Special number [on hold]

I have a question that is a little bit strange... What special characteristic does the number 69 have? and I'm not talking about the sexual connotation.
2
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2answers
60 views

Prove that $2^{9693}-1$ divisible by $7$

Prove that $2^{9693}-1$ divisible by $7$, by more than one way. my try... that, the power divisible by $3$ so it's divisible by $7$ like $2^3,2^6,2^{12}$ and I think it's wrong.
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1answer
62 views

Prove that: $\overbrace{222…222}$(repeated $1980$ times), divisible by $1982$ [on hold]

Prove that: $\overbrace{222...222}$(repeated $1980$ times), divisible by $1982$
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1answer
14 views

Convergence and Irrationality of $\frac{H_{(n,-n)}}{(n+1)^n}$ as $n$ approaches infinity

We define $H_{(a,b)}$ as the $a^{th}$ harmonic number of class $b$. In other words, $$H_{(a,b)}=\sum_{k=1}^a \frac{1}{k^b}$$ More information about generalized harmonic numbers can be found here. Let ...
2
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0answers
46 views

Finding Gaussian Primes along Lines in $\mathbb Z[i]$

I am trying to prove the following statement: For all positive integers $a$ does there exists a positive integer $b$ such that $a^2 + b^2$ is prime? (If so, can we provide such a $b$?) Given the ...
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0answers
9 views

Find all numbes $1\le a\le n-1$ which are prime to n and they are not witness Fermat of compositeness of n

Given the number $n=35$.Find all numbes $1\le a\le n-1$ which are prime to n and they are not witness Fermat of compositeness of n I found this problem on internet and i am trying to find a solution ...
1
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1answer
42 views

Book recomendations for learning number theory.

As mentioned in the title, i want to get started with number theory and I am searching for a book fitting my purposes. Ive just finished Calculus 1. I'd be glad if you could recommend a book written ...
0
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0answers
24 views

Sharper Bound for Minkowski's Convex Body Theorem

To satisfy the conditions of Minkowski's Convex Body Theorem we need a lattice $\Lambda$ with fundamental domain $T$ and a bounded convex symmetric subset of $\mathbb{R}^n$ (call it $X$) with ...
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0answers
56 views

Primes factors of n!

It seems that my question was not really clear so here is a new version : let $f(x,y)$ be the product of primes factors $\leq x$ of $y!$ let $g(x,y)$ be the product of primes factors $\gt x$ of $y!$ ...
2
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0answers
54 views

Does knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ sometimes offer a significant advantage in finding $r$?

Is there a cyclic group $\mathcal G$ with generator $g$ for which the discrete log problem is assumed to be hard, but knowing $g, g^r, g^{r^2}, g^{r^3}, \dotsc$ for random $r$ makes finding $r$ easy? ...
0
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0answers
29 views

Solve $x^n\equiv1 \pmod p$ for $x$ where $n$ is odd, $p$ prime [duplicate]

The solution to $x^3\equiv1 \pmod p$ has been discussed in Solve $x^3 \equiv 1 \pmod p$ for $x$ and explained elegantly by Arturo Magidin. The discussion established the form of $p$ and $x$. What ...
3
votes
1answer
111 views

Expected value when die is rolled $N$ times

Suppose we have a die with $K$ faces with numbers from 1 to $K$ written on it, and integers $L$ and $F$ ($0 < L \leq K$). We roll it $N$ times. Let $a_i$ be the number of times (out of the $N$ ...
1
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1answer
25 views

Find the smallest prime number that doesn't divide any 5-digit number whose digits are in strictly increasing order.

Find the smallest prime number that doesn't divide any 5-digit number whose digits are in strictly increasing order. I have posted an answer of my own below; any alternative solutions will also be ...
5
votes
2answers
51 views

Does the $5x + 1$ sequence for 7 reach a power of 2 or does it get stuck in a period?

This is much like $3x + 1$, except that if $x$ is odd, you do $5x + 1$. If $x = 7$, then we have 7, 36, 18, 9, 46, 23, 116, 58, 29, 146, 73, 366, 183, 916, 458, 229, ... I've iterated this twenty ...
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votes
2answers
58 views

Find all integer numbers $n$ such that $\frac{11n-5}{n+4}$ is a perfect square.

Find all integer numbers $n$, such that, $$\sqrt{\frac{11n-5}{n+4}}\in \mathbb{N}$$ I really tried but I couldn't guys, help please.
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1answer
36 views

The number of numbers lying between 1 and 200 which are divisible by either of 2 , 3 or 5?

The number of numbers lying between 1 and 200 which are divisible by either of two , three or five?
1
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3answers
185 views

Find the value of $x$ such that $\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4-x}}}}=x$

Find the value of $x$, $$\sqrt{4+\sqrt{4-\sqrt{4+\sqrt{4-x}}}}=x$$ Help guys please, I have tried and I got, $x=-2, x=1$, and I think it's wrong
2
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0answers
354 views

Does the following conjecture regarding the Riemann Zeta function hold?

Background We define the following numbers: $\phi_i$ is the i'th number such that it cannot be expressed as $\beta^n $ where $\beta $ is any positive integer and $n$ is an integer greater than 1. ...
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1answer
25 views

Number Theoretic Sum of three variables. Having trouble isolating two of them.

I encountered the following sum: $$\frac{c_{mn-p}}{c_0} = \sum_{d|p}\ \frac{a_{n-d}b_{m-\frac{p}{d}}}{a_0b_0} $$ Where $$a_i=0 \text{ when } i<0 \text{ and } b_j=0\text{ when } j<0\text{ and ...
5
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0answers
65 views

What primes were “pending” at the time of Wiles's proof of FLT?

I would like to know what instances of Fermat's Last Theorem were pending at the time of Wiles's proof. More specifically: what families of irregular primes had been discarded as possible ...
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0answers
24 views

Rabin's cryptography - when the message $M$ isn't coprime to $n = pq$

Say the message $M$ is a product of one of the primes $p$ or $q$, won't the $gcd$ of $M$ and $n$ (the public encryption key) give me $p$ or $q$? say $p = 11$ $q=19$ $n=11*19=209$ and $M=33$. ...
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1answer
25 views

Concept of combinations and number theory.

The number of six digit numbers of the form ababab(in base ten)each of which is a product of exactly six distinct primes?
3
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0answers
47 views

Approximate a large number with perfect powers

I'm dealing with number theory now and I have an interesting question. Every number can be approximated with two perfect powers, where perfect power is a number in form $$a^b$$$$a,b \geq 2, a,b \in ...
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0answers
23 views

Diophantine Equation with gcd. [duplicate]

Find all positive integers $a,b$ such that $\gcd(3^a+1,3^b+1)$ is a multiple of $ab$. I've given this problem many attempts but I can't seem to make any progress, there doesn't appear to be any way ...
-3
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1answer
48 views

solve $(x-3)^2 + (x+1)^2 + (4x-5)^2=0$ [on hold]

solve $(x-3)^2 + (x+1)^2 + (4x-5)^2=0$ this is what I have tried $$(x-3)^2=(x+1)^2=(4x-5)^2=0$$ $$x=3, x=-1, x=\frac{5}{4}$$
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0answers
53 views

How to compute $\mathbb Z_n \times \mathbb Z^*_m$? [on hold]

How to compute $\mathbb Z_n \times\mathbb Z_m^*$? (Here $\mathbb Z^*_m$ is the unit group mod $m$ and $(m,n)=1$.) In the paper Multiplicative properties of sets of residues it is said that by ...
1
vote
1answer
58 views

What the difference between the smallest two numbers from these numbers?

There are infinitely many integers $n$ bigger than $1$, such that if we divide $n$ by any integer $k$ where $2\leq k\leq 11$, the remainder is equal to $1$. What the difference between the smallest ...
2
votes
1answer
44 views

Determining parity of the multiplicative inverse?

Let $\mathbb{F}_p$ be a finite field of characteristic $p > 2$, for a fixed $p$. I will consider only prime fields, not $GF(p^n)$. Represent the $p$ elements of the field as integers $\{0,1,\ldots ...
6
votes
1answer
42 views

$F[[T]] \times F[[1/T]]$, fundamental domain.

Let $p$ be a prime number. Here is a link which shows how to see that $$(\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T)))/\mathbb{F}_p[T, 1/T]$$is compact using an adelic result. (Here $\mathbb{F}_p[T, ...
9
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0answers
80 views

$y^2 = x^3 - 26$, exist ideal satisfying conditions?

For the solution $(x, y) = (3, 1)$ of $y^2 = x^3 - 26$, does there necessarily exist an ideal $I$ of the integer ring $\mathbb{Z}[\sqrt{-26}]$ of $\mathbb{Q}(\sqrt{-26})$ such that $(y + ...
1
vote
1answer
14 views

Imprimitive Dirichlet Characters

I've started to read the fifth chapter of "Multiplicative Number Theory" by Harold Davenport and I got stuck at some point. Let me elaborate the part that i didn't quite understand. Let $\chi$ be ...
0
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0answers
29 views

For what $n$ can this sum be an integer? [duplicate]

Consider the well known $\sum_{k=1}^{n} \frac{1}{k}$ sum. My question is simple: How can we choose $n$ in order to make the sum integer? My approach: The first obvious solution is $n=1$. I tried ...
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0answers
31 views

How does Graham knows his number is really the upper bound to the dimension problem?

I know initially he stated that the answer is somewhere between 6 and Graham's number. How does he know that for Graham's number dimensions it is really impossible to color the lines that way? I know ...
3
votes
1answer
25 views
3
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2answers
117 views

Showing there is no triplet of positive integers $(a,b,c)$ satisfying $a^7+b^7=7^c$ [duplicate]

Show that $$a^7+b^7=7^c$$ has no positive integer solutions $(a,b,c)$. I've posted a general and way too long approach as an answer. How may one prove the claim more briefly and specifically?
2
votes
3answers
82 views

Is there always a square between two consecutive cubes?

Is there always a square between two consecutive cubes? I just thought of this question. It seems really simple and the answer is probably yes. Edit: I should have given this more than 2 seconds of ...
5
votes
2answers
73 views

$L$-function, easiest way to see the following sum?

What is the easiest way to see that$$\sum_{(m, n) \in \mathbb{Z}^2 \setminus \{0, 0\}} (m^2 + n^2)^{-s} = 4\zeta(s)L(s, \chi)?$$Here $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to ...
5
votes
1answer
74 views
+100

Extending 2-adic valuation to real numbers

When proving Monsky's theorem, one of the steps, which, from what I have so far seen, no proof can avoid, is extending the 2-adic valuation to all real numbers, so that it still satisfies ...
1
vote
1answer
62 views

Expansion of factorial of a natural number as a summation

Factorial of any natural number $n$ (i.e. $n\in N$) can be expanded as a summation $$n!=1+\sum_{i=1}^{n-1}(i\times i!)$$$$=1+1\times 1!+2\times 2!+3\times 3!+4\times 4!+..............+(n-1)\times ...
2
votes
1answer
44 views

Fermat's Theorem and primitive $n$th roots of unity

Fermat's little theorem states that if $p$ is a prime number, then for any integer $a$ such that $p \not \mid a$ we have that $a^{p − 1} \equiv 1 \operatorname{mod} p$. Suppose $p =17$, then we know ...