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.

learn more… | top users | synonyms (1)

3
votes
4answers
122 views

How to calculate $9^{47^{51}} \mod 67$?

I've looked at some other related things on here, but this seems a little more complicated with the double exponentiation. Is there a general algorithm to calculate $a^{c_1^{c_2^{...^{c_n}}}} \mod p$ ...
1
vote
1answer
56 views

Prove that a Fermat Number cannot be a Carmichael Number?

Prove that a Fermat Number cannot be a Carmichael number: Fermat numbers are of the form $2^{2^n}+1$ and $F$($n$) denotes the $n$th Fermat number. If $F$($n$) is a Carmichael number, that would mean ...
0
votes
2answers
128 views

Prove *directly*: Even perfect squares have even square roots.

Is it possible to prove directly that even perfect squares have even square roots? Or, symbolically: $\forall n \in \mathbb{Z},\ \ n^2 \text{ is even } \Rightarrow n \text{ is even }$ The indirect ...
1
vote
1answer
85 views

Counting the number of partitions

Let $P$ be a set of $7$ different prime numbers and $C$ a set of $28$ different composite numbers each of which is a product of two (not necessarily different) numbers from $P$. The set $C$ is divided ...
1
vote
0answers
38 views

Hasse-Weil bound for even degree polynomial

Consider the equation $y^2 = f(x)$, in the finite field $\mathbb{F}_q$($q$ is a prime power) where $f(\cdot)$ is a monic polynoimal of even degree (greater than or equal to $4$) with integer ...
5
votes
1answer
95 views

There are at least two solutions such that $2p_n=p_a+p_b$ ($p$ being prime)

I've stumbled across this playing around and summing primes at random during a boring lecture. Is this a known conjecture? Can it be proven? My conjecture: There exists at least one non trivial ...
1
vote
0answers
37 views

Given some arbitrary roots of a polynomial p(x,y,z,…) with integer coefficients, is it possible to tell if p has a root in the Gaussian integers?

I'm trying to find if p(x,y,z,...)=0 has a Gaussian integer root (more specifically, I want to find if p has a Gaussian integer root where the imaginary components are even, but if that can't be done, ...
5
votes
4answers
112 views

Prove that $\frac{2^a+1}{2^b-1}$ is not an integer

Let $a$ and $b$ be positive integers with $a>b>2$. Prove that $\frac{2^a+1}{2^b-1}$ is not an integer. This is equivalent to showing there always exists some power of a prime $p$ such that $2^...
7
votes
1answer
106 views

Why can't negative numbers be prime? [duplicate]

I was in a lecture on primes, when it occurred to me what negative numbers are excluded. Why exactly?
1
vote
0answers
134 views

solve $x^y-y^x=xy^2-19,$ $x,y\in\mathbb{Z}$

I have been struggling to solve this exercise but with no result: $$x^y-y^x=xy^2-19,$$ $x,y\in{\mathbb Z}$ I have started to think it has no solutions at all. I have no idea how to solve it so I was ...
0
votes
1answer
68 views

Find the Smallest Value

Find the smallest value of $$a + \frac {1}{(a-b)b} $$ where a>b>0 I found this question in AM-GM inequality problems but I am stuck at this
0
votes
1answer
40 views

How many $N$ digits binary numbers can be formed where $01$ exactly $k$ times is repeated.

How many $N$ digits binary numbers can be formed where $01$ exactly $k$ times is repeated. Note: first digit can't be zero.
2
votes
0answers
23 views

Algorithm similar to the Euclidean algorithm

Given a prime number $p$, and an initial number $1<a<p$, what would be the upper bound on the iteration number of the following algorithm? (1) if $a=1$ then stop (2) else replace $a$ by $p - a\...
2
votes
0answers
60 views

Prime elements of ring $\mathbb{Z}[\sqrt{-21}]$ [closed]

Find prime elements of the ring $\mathbb{Z}[\sqrt {-21}]$. Please help with some ideas.
0
votes
0answers
20 views

A question about approximation by fractions

I want to solve one of Rosen Elementary Number Theory problems; I had a problem during proving one of them, here is the question: Show that if $\frac{p_k}{q_{k}} $and $\frac{p_{k+1}}{q_{k+1}}$ are ...
5
votes
1answer
53 views

Is it true that $x \equiv 1 \pmod{p}$?

If $x^{m} \equiv 1 \pmod{p}$ for some prime $p$ with $\gcd(m,p-1) =1$, is it true that $x \equiv 1 \pmod{p}$? I was wondering about this result and it seems that since the maximal order of $x$, $p-1$...
5
votes
3answers
83 views

Sum of three consecutive prime numbers is $173$

If I tell you that the sum of three consecutive prime numbers is $173$, how quickly could we find the biggest of these numbers?
0
votes
0answers
48 views

To solve multivariate polynomial equations

For a system of multivariate polynomial equations like this: $$ \left( {\begin{array}{*{20}c} {\frac{{124}} {3}} & { - 24} & {\frac{{ - 68}} {3}} & {\frac{{68}} {3}} \\ {32} & {...
0
votes
1answer
38 views

Upper and Lower Bound on Partition Function

The partition function $p(n)$ counts the number of ways an integer can be expressed as a sum. For example, $p(4)=5$ as $$4=3+1=2+2=2+1+1=1+1+1+1$$ Hardy and Ramanujan were able to develop a converging ...
8
votes
0answers
93 views

Generalizing the growth of sums of two squares

Consider the set $S$ of numbers which are the sum of two (integer) squares, and define $S(n)$ as the number of members of $S$ in $\{1,2,\ldots,n\}.$ It is well-known that $$ S(n) \sim \frac{Kn}{\sqrt{\...
2
votes
1answer
38 views

Algorithm for finding the representation of an integer as a sum of two squares

We know that an integer $n$ is the sum of two squares if and only if all its prime divisors $p$ of the form $p \equiv 3 \pmod4$ have an even exponent in the prime factor decomposition of $n$. My ...
1
vote
1answer
38 views

is the sevenacci sequence the only generalization of the fibonacci sequence that has no odd prime terms?

This question came up recently. Obviously the answer is no, but how can I know for sure without checking a whole bunch of generalizations? $s_1 = ... = s_6 = 0, s_7 = 1, s_n = s_{n-1} + ... + s_{n-...
3
votes
2answers
46 views

Estimate for $\sum_{q=1}^{M}\frac{\varphi(q)}{q^{2}}$ Related to Bourgain Paper [duplicate]

Let $N\gg 1$ be a large parameter, which I ultimately want to let tend to infinity. I am reading an old paper of Bourgain, where he claims the lower bound (Equation 2.50, pg. 118) $$\sum_{q=1}^{N^{1/...
12
votes
0answers
154 views

How did Hecke come up with Hecke-operators?

I'm currently studying Hecke-operators and I'm curious how Hecke came up with them. The original definition he gave in his paper is $$\left( f \mid T_n\right) (z) = n^{k - 1} \sum_{ad = n, \, b \mod d,...
1
vote
0answers
34 views

If $p$ is unramified in every subfield of $K$, does it mean $p$ is unramified in $K$?

I am wondering if $p$ being unramified in every subfield of $K$ means $p$ is unramified in $K$. Any hints?
8
votes
2answers
177 views

Prove that for some $x, y \in \mathbb{Z}^+$, if $(x-1)(y-1), xy, (x+1)(y+1)$ are all squares then $x = y$.

Prove that for some $x, y \in \mathbb{Z}^+$, if $(x-1)(y-1), xy, (x+1)(y+1)$ are all squares then $x = y$. I tried taking all possible combinations $\bmod 3$ and $\bmod 4$ and it has a solution only ...
6
votes
1answer
100 views

Restricted equality involving prime numbers

Given three real numbers such that $a + b + c = 0$, it can be proved that \begin{align*} \frac{a^{5} + b^{5} + c^{5}}{5} & = \frac{a^{3} + b^{3} + c^{3}}{3}\cdot \frac{a^{2} + b^{2} + c^{2}}{2}\\ \...
1
vote
1answer
49 views

Why does this method for finding the number of factors for number X not work?

As you may know, in order to find the number of factors for natural number X, we take the prime factorization, add one to each exponent, and multiply, as such. $...
3
votes
3answers
62 views

Let $a,b,x,y\in \mathbb{Z}$, with $ax-by=1$. How to prove that $gcd(a+b,x+y)=1$?

My guess is that if i start with this $ax-by=1$, by transforming that into a $gcd()$ form, like $gcd(ax,-by)$, or $gcd(a,-b)$ and this way i could be able to reach the end result. However, i can't ...
3
votes
0answers
63 views

Smallest number $m$ with $gnu(m)=2017\ $?

$gnu(n)$ denotes the number of groups of order $n$ upto isomorphy. $moa(n)$ denotes the smallest number $m$ with $gnu(m)=n$ $$m=259,083,319,343,897,905=5\cdot 2011\cdot 24133\cdot 1067692187$$ ...
0
votes
2answers
36 views

The sum of the perimeter of regular polygons inscribed inside of regular polygons

This is a question combining number theory and geometry. I am asking it purely from curiosity, but I think it might be a useful and interesting question. Start with an equilateral triangle of ...
1
vote
1answer
48 views

Codewars “Double Cola” Brainteaser

I encountered this brainteaser while coding, but it is essentially a math problem: Sheldon, Leonard, Penny, Rajesh and Howard are in the queue for a "Double Cola" drink vending machine; there ...
1
vote
0answers
24 views

How to show that $\frac{((1+T)^{p^n-1}-1)}{((1+T)^{p^e-1}-1)}$ is a polynomial?

I am currently stuck at computing the polynomial (with $0<e<n$): $$\frac{((1+T)^{p^n-1}-1)}{((1+T)^{p^e-1}-1)}$$ From the context I already know that it should be a distinguished polynomial, as ...
16
votes
1answer
1k views

Failure of an elementary 'proof' of Fermat's Last Theorem?

Can someone explain to me why this does not constitute a proof of Fermat's Last Theorem, please? Basically, using something I've read online, it appears you can write an equation for $(a, b, c)$ to ...
17
votes
0answers
236 views

Finding primes so that $x^p+y^p=z^p$ is unsolvable in the p-adic units

On my number theory exam yesterday, we had the following interesting problem related to Fermat's last theorem: Suppose $p>2$ is a prime. Show that $x^p+y^p=z^p$ has a solution in $\mathbb{Z}_p^{\...
3
votes
3answers
200 views

$S_n$ is an integer for all integers $n$

Let $a$ be a non-zero real number. For each integer $n$, we define $S_n = a^n + a^{-n}$. Prove that if for some integer $k$, the sums $S_k$ and $S_{k+1}$ are integers, then the sums $S_n$ are integers ...
1
vote
4answers
131 views

Suppose that $a$ and $b$ satisfy $a^2b|a^3+b^3$. Prove that $a=b$.

Suppose that $a$ and $b \in \mathbb{Z}^+$ satisfy $a^2b|a^3+b^3$. Prove that $a=b$. I have reduced the above formulation to these two cases. Assuming $b = a + k$. Proving that any of the below two ...
-2
votes
1answer
30 views

find the number of tuples of positive integers [closed]

find the number of tuples (a,b,c,d) of positive integers \begin{array}{l} {a^3} = {b^2}\\ {c^3} = {d^2}\\ c - a = 64 \end{array} answer should be one of 0 , 1 , 2 , 4
0
votes
0answers
11 views

On a Corollary of Liouville's Theorem

I want to prove, using Liouville's Theorem that: Let $\theta$ be an irrational algebraic number of degree $n$. Then, given any $\epsilon > 0$ there exist only a finite number of pairs of integers $...
2
votes
2answers
27 views

Show that two numbers divided by their GCD are coprime

Let $a, b \in \mathbb{Z} \setminus \{0\}$ and $d = gcd(a, b)$. Show that $gcd(\frac{a}{d}, \frac{b}{d}) = 1$. I tried proving this by contradiction and showing that otherwise $d$ isn't the gcd of $a$...
0
votes
1answer
29 views

number theory problem finding triplets [closed]

Find number of triplets of positive integers satisfying $2^a-5^b\cdot 7^c=1$ Given options are $0 , 1 , 2$ or infinite.
0
votes
0answers
11 views

When does the equality hold in Dias da Silva - Hamidoune Theorem?

Let $p$ be prime number and let $A$ be a $k$-elements subset of $\mathbb{Z}/p\mathbb{Z}$. Dias da Silva - Hamidoune Theorem states that $|h^{\hat{}}A| \geq \min(p, hk -h^2 + 1)$, where $h$ is an ...
0
votes
2answers
48 views

Is $8x+5$ always square non-residue modulo $8x^2+7$ for natural $x$?

Is $8x+5$ always square non-residue modulo $8x^2+7$ for natural $x$? This holds for $x$ up to $10^8$. The kronecker symbol is never $1$ also. Working modulo $8$ doesn't appear straightforward since $...
1
vote
2answers
44 views

3 digit Prime Palindrome Numbers.

Question. How many three digit palindrome number are prime? Ans. Any 3 digit palindrome number is of type "aba" where b can be chosen from the numbers 0 to 9 and a can be chosen from 1 to 9. So the ...
2
votes
2answers
41 views

Logarithm of >2 numbers

I am learning logarithms and i found that $log(a*b) = log(a)+log(b)$ I tried to apply the same principle for three numbers like $log(a*b*c) = log(a)+log(b)+log(c)$ but it didn't work as i expected. ...
1
vote
4answers
81 views

If $x$ and $y$ are non-negative integers for which $(xy-7)^2=x^2+y^2$. Find the sum of all possible values of $x$.

I am not able to reach to the answer. I have used discriminant as $x$ and $y$ are both integers but it didn't give any hint to reach to answer. I am not able to understand how should I deal with these ...
0
votes
0answers
29 views

Probability distribution of $\omega'(n)$. [duplicate]

$\omega(n)$ is the number of distinct prime factors of $n$ and $\omega'(n)$ is the number of distinct prime factors of $n$ with multiplicity. For example if $p,q$ are prime numbers then $\omega(p^2q)=...
2
votes
1answer
50 views

Five exponentials theorem

The six exponentials theorem is proved in most textbooks on transcendental number theory, and the four exponent conjecture is an open problem. Is there any good/accessible exposition of the five ...
0
votes
0answers
27 views

Deducing Lindemann-Weierstrass from Baker's theorem

I'm aware that Baker's theorem with $n=1$ (for one algebraic number only) follows from that of Lindemann-Weierstrass. It is also often mentioned that Baker's result is a generalization of Lindemann-...
0
votes
2answers
68 views

Find the first digit of a number

I have seen this question but i could not find any answers.Let A= a*b*c*d.... very huge multiplication So what we can do take ...