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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4
votes
2answers
46 views

Relation $S(2x)=2S(x)-9N(x)$.

Let $S(x)$ be the sum of digits of number $x$ and $N(x)$ be the number of digits of $x$ greater than $4$. Prove that $S(2x)=2S(x)-9N(x)$. For example, if $x=1992$ then $S(x)=1+9+9+2=21$ and ...
10
votes
3answers
377 views

BMO2 2016 Number Theory Problem

Suppose that $p$ is a prime number and that there are different positive integers $u$ and $v$ such that $p^2$ is the mean of $u^2$ and $v^2$. Prove that $2p−u−v$ is a square or twice a square. Can ...
1
vote
1answer
38 views

estimation for n-th prime

The famous theorem of Hadamard and Vallee-Poussin https://en.wikipedia.org/wiki/Prime_number_theorem implies that $p_n\sim n\ln n$, so $C_1 n\ln n \le p_n \le C_2 n\ln n$ holds for all $n\ge 2$ with ...
3
votes
2answers
61 views

Largest number of consecutive positive integers whose sum is exactly $2014$.

$97+98+ ...........+114+115 = 2014$. Here sum of $19$ consecutive numbers is $2014$. Find the largest number of consecutive positive integers whose sum is exactly 2014 and justify why you think ...
8
votes
3answers
417 views

Find all values x, y and z which satisfy the equation $(x^2 + 1)(y^2 + 1) = z^2 + 1$ given that $(x^2 + 1)$ and $(y^2 + 1)$ are both primes.

Find all positive integers x, y, z which satisfy the equation $(x^2 + 1)(y^2 + 1) = z^2 + 1$ given that $(x^2 + 1)$ and $(y^2 + 1)$ are both primes. It seems trivial that the only set of integers x, ...
-1
votes
0answers
111 views

About a Sequence of Prime Numbers inspired by the Green Tao Theorem

I am learning math so this question may seem obvious. It is known from the Green Tao theorem that the sequence of prime numbers contains arbitrarily long arithmetic progressions. The Green Tao theorem ...
1
vote
0answers
32 views

Different representation of $f(n) = \sum_{d|n; \ \sqrt n\le d \le n}(-1)^d$

I am looking for a different way to calculate the following sum where $d,n\in \mathbb N$: $$f(n) = \sum_{d|n; \ \sqrt n\le d \le n}(-1)^d$$ Here are some example results for different values of n ...
8
votes
0answers
171 views
+100

Modular transformation of $\eta(\tau)$

I know that the Dedekind $\eta$ function can be represented in the form$$\eta(\tau) = q^{1\over{24}} \prod_{n = 1}^\infty (1 - q^n) = \sum_{n = -\infty}^\infty (-1)^n q^{{3\over2}\left(n - ...
1
vote
1answer
48 views

Find all functions $f:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ [closed]

Find all functions $f:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$, such that $i)f(a,a)=a$ $ii)f(a,b)=f(b,a)$ $iii)\mbox{If } a>b, \mbox{then } f(a,b)=\dfrac{a}{a-b}f(a-b,b)$ For the form of the ...
1
vote
0answers
38 views

Non trivial results in graph theory/combinatorics coming from number theory

Are there any non-trivial results in graph theory that can be deduced from number theory or arithmetic geometry? I am not looking for expander graphs or applications however I am looking for ...
4
votes
2answers
72 views

When is $\bigl( \frac{a}{b} \bigr)^{3} \pm \bigl( \frac{x}{y}\bigr)^{3}$ an integer?

I am trying solve this form, but it appears not easy problem, and also I can't find references about it. I suppose some constrains should be stated, like "$b,y > 1$" and "$\gcd(a,b) \gcd(x/y) = ...
2
votes
1answer
89 views

Explicit form for $\sum_p \ln(\ln(p))$?

Riemann gave an explicit form for the counting function of the primes. Is there an explicit form for the counting function $f(x) = \sum_p \ln(\ln(p))$ where the sum is over $p$ : the number of primes ...
1
vote
0answers
20 views

Euclids Lemma p|abc

I'm hoping yall can let me know if this proof looks okay. I'm trying to prove "If p|abc then p|a or p|b or p|c" This is what I came up with for the proof:
1
vote
1answer
48 views

Prove that $L_n = \alpha^n +\beta^n$ for all integers $n\geq 0$

Let $\alpha =\left(\frac{1+\sqrt{5}}{2}\right)$ and $\beta = \left(\frac{1-\sqrt{5}}{2}\right)$. Prove that $L_n = \alpha^n +\beta^n$ for all integers $n\geq 0$ where $L_n$ denotes the Lucas numbers. ...
0
votes
0answers
148 views
+50

Reformulation of Goldbach's Conjecture as optimization problem correct?

Question I think I managed to reformulate a stronger version of Goldbach's conjecture as an optimization problem: $$ \frac{\partial F_n}{\partial a_n} = \frac{\partial F_n}{\partial \overline a_n} ...
0
votes
2answers
28 views

Let $F_n$ denote the nth Fibonacci number and prove that the following re true for every possible integer $n$

$$\sum_{i = 1}^n F_{i}^2 = F_n F_{n+1}$$ -I solved a similar Fibonacci sequence that was the following: $$\sum_{i = 1}^n F_i = F_{n + 2} - 1$$ But, I am having trouble with this one, any help is ...
0
votes
0answers
21 views

Linear equations and the gcd

given integers $a$ & $b$ it can be shown that the equation: $$ax + by = gcd(a,b)$$ always has a solution. To prove this we assume the exitence of two solutions $$(x_1, y_1)$$ & $$(x_2, y_2)$$ ...
1
vote
1answer
47 views

Prove that there doesn't exist prime numbers $a, b, c$ s.t. $a^2=b^2+c^3$

I first showed that if $a,b,c \neq$ 2, then they are odd and therefore are never equal. Then I consider the cases where $a=2$, $b=2$ and $c=2$. It seems to be unnecessarily long so is there a more ...
3
votes
3answers
848 views

Prove that the sequence $\cos(n\pi/3)$ does not converge

EDIT: Using a rigorous formal proof, I need to prove that this sequence does not diverge. I of course understand why it doesn't converge... $n=1$ to infinity of course. So, I have a bit of trouble ...
1
vote
4answers
58 views

Prove that $f(n) = 3n^5 + 5n^3 + 7n$ is divisible by 15 for every integer $n$

So far I have only been able to complete the base case for which I got the following: $$f(n) = 3n^5 + 5n^3 + 7n$$ $$f(n) = 3(1)^5 = 5(1)^3 + 7(1)$$ $$f(n) = 3 + 5 + 7$$ $$15/15 = 1$$ From here ...
0
votes
0answers
39 views

Regarding the solvable congruent numbers from the congruent number problem is this algorithm efficient?

The congruent number problem states that if rational x,y,z such that $x^2+y^2==z^2$ then $N=x y/2$ is a congruent number if N is an integer. If N isn't an integers x,y,z can be scaled up until it is. ...
2
votes
1answer
25 views

$ord_p(x)$ -Units and Irreducibles

Let $p$ be a prime number in $\mathbb{Z}$. Let $R = R_p = \{x \in \mathbb{Q}\ |\ ord_p(x)\geq0\}$, which is a subring of $\mathbb{Q}$. (a) Show that $x \in R$ is a unit iff ...
1
vote
1answer
13 views
2
votes
2answers
258 views

Milton Green's lower bounds of the busy beaver function

Wikipedia states that Milton Green demonstrated in 1964, that the busy beaver function $\Sigma(n)$ has the lower bound $$\Sigma(2k)>3\uparrow^{k-2}3$$ I read the talk about the busy beaver ...
2
votes
1answer
44 views

Summation of harmonic series using Bertrand's Postulate

Question: Let $n\ge 2$ be an integer. Show that $$1+\frac 12+\frac 13+\dots+\frac 1n$$ is not an integer. It is a well-known question, even I have answered similar questions several times in ...
0
votes
0answers
13 views

How the sieving part of quadratic sieve actually works?

I am trying to implement quadratic sieve algorithm as it's described in wiki. I understand most of it, except the part of the sieving example. In the example they use $N = 15347$ with base prime ...
1
vote
2answers
60 views

$11^{-1}$ modulo $91$ is $58$. Why?

I am reading wiki article about Quadratic Sieve and it says $11^{-1}$ modulo $91$ is $58$ Why? How is it been calculated?
1
vote
1answer
48 views

A conjecture about quadratic residues given $p \equiv 5 \pmod 8$

Original Problem $p$ is a prime that is congruent to $5$ modulo $8$ and $a$ is a quadratic residue modulo $p$. Prove that excactly one of $x_1=a^{\frac{p+3}{8}},x_2=(2a)(4a)^{\frac{p-5}{8}}$ is the ...
0
votes
2answers
38 views

question about number theory [duplicate]

If $p$ is a prime number and $p\mid a^p-b^p$. Then $p^2\mid a^p -b^p$.
2
votes
0answers
42 views

On the sum of the reciprocals of the zeros of $\zeta(s)$

It is well known that whenever $\rho$ is a nontrivial zero of the Riemann zeta function $\zeta(s)$, then $1-\rho$ is also a zero. But does the equality $\Re \sum_{\rho} \dfrac{1}{\rho} = \Re ...
16
votes
2answers
2k views

Can every even integer be expressed as the difference of two primes?

Can every even integer be expressed as the difference of two primes? If so, is there any elementary proof?
1
vote
1answer
47 views

Are there prime gaps of every size?

Is it true that for every even natural number $k$ there exists some $n \in \mathbb{N}$ such that $g_n = p_{n+1} - p_n = k$? I don't know how to approach the problem at all, and in fact I don't even ...
2
votes
1answer
39 views

What's so special about hyperbolic curves?

This is really a two-part question, but I would be happy to get an answer for either bit. By a hyperbolic curve as defined by e.g. Szamuely in Galois Groups and Fundamental Groups (p.137) I mean an ...
1
vote
2answers
54 views

Solving $12x \equiv 20 \pmod{38}$

$12x \equiv 20 \pmod{38}$ $gcd(12,38)=2$ using Euclidean Algorithm. There is a solution since $2|20$. Use the Extended Euclidean Algorithm $2=12*-3 +1*38$ Then $20=2*10=12*-3*10+1*38*10$ so ...
1
vote
0answers
59 views

Formula for this pattern? [closed]

I have this pattern : 1, 1, 5, 9, 25, 54, 131, 295, 691, 1579, 3655, 8396, 19365 Is there an explicit way to express the nth term for this sequence of numbers ? Can´t find any pattern on this.
5
votes
1answer
43 views

Modular Arithmetic - summing from 1 to a prime

Apologises for the vague title; I couldn't think of anything better to call it. I'm currently working on the following question: Consider the equation $\sum_{i=1}^{5} \frac{1}{i} = \frac{X}{5Y}$. ...
1
vote
5answers
68 views

How connect $x^2+xy+y^2$ to $j^3*4*n-27 = t^2$

$x^2 + xy + y^2 = (x^3 - y^3)/(x - y)$ Now let me show a subject not connected with above form (at least in some known way). By trying solve equation $1 \cdot 4 \cdot n - 27 = t^2$ ($n,t$ ...
2
votes
1answer
101 views

“The PNT obtained by statistical methods”

In a famous book "What is Mathematics" by Richard Courant, Herbert Robbins the authors presented not a rigorous proof, but "an argument that at least makes plausible the truth of Gauss's famous ...
3
votes
2answers
58 views

Is the last digit of this number :$ {{4^4}^n}+1 $ always $7 $ for $n>1$ and could this be prime?

Some computations in wolfram alpha for $n=2,3,4,5 ,6$ showed that the last digit of this number $ {{4^4}^n}+1 $ for $n>1$ always $7$ . My question here :How do I know if it's last digit always ...
377
votes
28answers
42k views

Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ...
27
votes
2answers
699 views

Fibonacci $\equiv -1 \mod p^2$

Is there a prime $p > 3$ such that the Fibonacci number $F_{np} \equiv -1 \mod p^2$ for some natural number $n$? I know none of the first $1000$ primes $> 3$ qualify. EDIT: In response to ...
1
vote
2answers
28 views

Show that $c^2 \equiv -1 \pmod p$

Let $p \equiv 1 \pmod4$ be a prime. Write $p$ in the form $p=a^2+b^2$ where $a$ and $b$ are integers. Let $c \equiv ab^{-1} \pmod p$. Show that $c^2 \equiv -1 \pmod p$ $p = 4n+1$ where $n$ is an ...
7
votes
2answers
284 views

Prove Divisibility In Fibonacci Sequence Over A Prime Number

In The Fibonacci sequence which is defined as $$ F_n=F_{n-1}+F_{n-2}, $$ lets say we have the number $p$ which is an odd prime. Prove that: $F_{p-1} + F_{p+1} -1$ Is divisible by $p$. Prove that ...
1
vote
1answer
24 views

Dense on the unit circle

I am reading: "It is sufficient to show that the points $z_n = e^{2\pi in \xi}$ $\:\:n = (1, 2, 3...)$ are dense on the unit circle. ( $\xi$ is an irrational number)" How is this possible? Can ...
0
votes
0answers
61 views

An analytic formula for the sum of the logs of primes.

I just read in Martin Klazar's Intoduction to Number Theory (page 53), that $\sum_{p\leq x} \log p - \log (p-1) = \log\log x + \gamma + O(1/\log x)$. Where $\gamma$ is the Euler-Mascheroni constant, ...
1
vote
0answers
83 views

Existence of a $G(x)$ that can generate all the even numbers?

Question This is a "spin-off" question of: Reformulation of Goldbach's Conjecture as optimization problem correct? I was wondering if a function existed such that: $$ G(x)^2 = ...
-5
votes
0answers
60 views

Finding Largest Number [closed]

Find largest number smaller than N with exactly K number of divisors. K is upto $40$. This means our final number doesn't contain more than $5$ unique prime divisors. N ranges upto $2^{40}$
4
votes
1answer
79 views

The first $4$ primes $p$ for which $15347$ has a square root mod $p$ are $2, 17, 23,$ and $29$

I am reading about Quadratic Sieve article in wiki and I don't understand the sieve part. The article says: The first $4$ primes $p$ for which $15347$ has a square root mod $p$ are $2, 17, 23,$ ...
1
vote
1answer
30 views

$C(n)+P(n)+S(n)$ always composites?

Let $C(n)$ be the concatenation of first $n$ primes, let $P(n)$ be the product of first $n$ primes, and let $S(n)$ be the sum of the first $n$ primes. It is not surprising that $C(n) - P(n) - S(n)$ is ...
5
votes
2answers
233 views

On odd perfect numbers $N$ given in the Eulerian form $N = {q^k}{n^2}$

Note: This question was cross-posted from MO. Preamble: I apologize in advance if this particular MSE post would appear to be a bit of a polymath approach, I just had to put down all the details to ...