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)

8
votes
6answers
346 views

Finding the positive integer numbers to get $\frac{\pi ^2}{9}$

As we know, there are many formulas of $\pi$ , one of them $$\frac{\pi ^2}{6}=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}...... $$ and this $$\frac{\pi ^2}{8}=\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}......
2
votes
1answer
50 views

Linear independence of Eisenstein Space Basis

I'm currently working throught Diamond and Shurman's book "A First Course on Modular Forms". On page 129 Theorem 4.5.2 states Let $ N $ be a positive integer and let $ k \geq 3 $. The set $$ \...
2
votes
2answers
23 views

Prove that there exist $i \neq j$ with $ia_i \equiv ja_j$ mod $p$.

Let $p$ be an odd prime, and assume that $a_i \equiv i \pmod p$ for $ 1 \leq i \leq p-1$. Prove that there exist $i \neq j$ with $ia_i \equiv ja_j \pmod p$. This is one of my textbook problem but I ...
1
vote
1answer
393 views

In an examination the maximum marks for each of the three papers are 50 each. Maximum marks for the fourth paper are 100. …

Problem : In an examination the maximum marks for each of the three papers are 50 each. Maximum marks for the fourth paper are 100. Find the number of ways in which the candidate can score 60 % marks ...
1
vote
1answer
68 views

Confusion on Inert Primes in Ireland and Rosen

In Ireland and Rosen, the following law for inert rational primes in a quadratic field is stated as: if $p\nmid \delta_K$, where $\delta_K$ is the discriminant of the quadratic field, and $d$ is a ...
1
vote
0answers
22 views

Need help applying euclidean algorithm to polynomials

I don't understand how the euclidean algorithm can be used to find the gcd of two polynomials in the following example: $x^3-27=(x-3)(x^2+3x+9)$ $2x^3-11x^2+16x-3=(x-3)(2x^2-5x+1)$ so we expect the ...
2
votes
0answers
214 views

Who first proved Fermat's Last Theorem for polynomials and when?

Who first proved Fermat's Last Theorem for polynomials and when? I have a proof using the Mason-Stothers Theroem, but the result is much older. Does anyone know the original proof or prover? Or at ...
1
vote
2answers
60 views

Can two pythagoras triplet have a common number

If I have a pythagoras triplet $(a,b,c)$ such that $$a^2+b^2=c^2$$ then is there another triplet $(a,d,e)$ possible such that $$a^2+d^2=e^2, \; b\neq d$$
2
votes
1answer
95 views

The exponential extension of $\mathbb{Q}$ is a proper subset of $\mathbb{C}$?

This question come from a recent post Exponential extension of $\mathbb{Q}$. An exponential field is a field $\mathbb{K}$ where it's well defined a function $E:\mathbb{K} \rightarrow \mathbb{K}$ ...
5
votes
1answer
294 views

At which p-adic fields does the equation have no solution?

I have to check if the equation $3x^2+5y^2-7z^2=0$ has a non-trivial solution in $\mathbb{Q}$. If it has, I have to find at least one. If it doesn't have, I have to find at which p-adic fields it has ...
2
votes
0answers
111 views

What is the relationship between GRH and Goldbach Conjecture?

We know that we can prove weak Goldbach Conjecture (three prime theorem) if we assume GRH (Hardy-Littlewood had proved this). Can we also prove strong Goldbach Conjecture if we assume GRH ? Also, ...
0
votes
1answer
25 views

Points where this three varible function takes the value $1$

I need to find minima of this function. $f(a,b,c)=2^a-5^b\cdot7^c$ where $a,b,c$ are positive integer I need to prove that for any value of a,b,c the value of function can never be 1. Tried ...
0
votes
1answer
205 views

If r,s,t are prime numbers and p,q are the positive integers such that LCM of p,q is $r^2s^4t^2$ then find the …

Problem : If r,s,t are prime numbers and p,q are the positive integers such that LCM of p,q is $r^2s^4t^2$ then find the number of ordered pairs (p,q)? Can we use this : let $r^2s^4t^2$ = $2^23^45^2$...
0
votes
1answer
40 views

Available in 6-packs, 9-packs, 20-packs

Started with an algorithms problem which says item is sold in 3 different sizes of boxes. These 3 boxes have 6, 9, 20 items each. Input is n, figure out if you can ...
3
votes
0answers
40 views

Is this “by symmetry” statement valid?

Problem: Let $p,q,r$ be integers such that $\gcd(p,q,r)=1$. Prove that there exists an integer $A$ such that $\gcd(p,q+Ar)=1$. A start: Assume for the sake of contradiction that $\gcd(p,q+Ar)>1$ ...
2
votes
1answer
36 views

Diophantine Equation 1

I want to solve for positive integral values of $x$ and $y$: $$1216562x=87654321y+a$$ Here $a$ is a positive integer. For example if $a=40642509$ then one solution is : $x=37716$ and $y=523$ How do I ...
3
votes
1answer
484 views

Kummer-Dedekind's factorisation theorem

For a number field extension $K$ of $\mathbb{Q}$ one can factorise almost all prime ideals $(p)$ in the extension $K$, except finitely many, easily by factorising minimal polynomials in finite fields....
3
votes
3answers
52 views

divisibility of $n^{15} - n^3$ by $32760$

I have a question & I have no idea where to begin. I hope someone here can help me. Been stuck for a while. Prove or disprove: $n^{15} - n^3$ is divisible by $32760$ for all $n \ge 0$.
0
votes
1answer
48 views

Extended Euclid Algorithm [duplicate]

A Linear Diophantine Equation is of the following form: $Ax+By+C=0$, where $x_1\le x\le x_2$ and $y_1\le y\le y_2$. If the value of $A$, $B$, $C$, $x_1$, $x_2$, $y_1$, $y_2$ are given and $x_1\le x_2$...
1
vote
0answers
24 views

Problem on solving congruence equation 2

I want to find smallest solution $x$ for the following equation for known positive integer values $a$, $b$ and $m$. $$a-bx+2^{6x+1}\equiv0\ \pmod{m}$$ Any help will be appreciated.
6
votes
2answers
159 views

Primes for which a polynomial splits completely

Suppose that $f(x) \in \mathbb{Z}[x]$ is an irreducible polynomial over $\mathbb{Q}$. Nevertheless, it may be the case that $f(x)$ is reducible modulo $p$ for some prime $p$. What is the density of ...
7
votes
2answers
82 views

Show that $\dfrac{2^p}{p}$ has remainder of $2$ for any prime $p \geq 3$

A bonus question on my last math exam I haven't been able to solve. Thanks for the help.
0
votes
1answer
111 views

$\pi$ normal to the base $10$ [closed]

If $\pi$ is normal to base $10$, why would we expect to find a string of ten $0$'s in its decimal expansion?
2
votes
1answer
104 views

Is there an Asymptotic Formula for the Largest Prime Factor of a Number?

It seems the asymptotic formula, $\sum_{n=2}^{x} P(n)$ ~ $\frac{\pi^2}{12}\frac{x^{2}}{log(x)}$ as $x \rightarrow \infty$ where $P(n)$ is the largest prime factor of the positive integer $n$, cannot ...
2
votes
2answers
1k views

Are square numbers also known as rectangular numbers, too?

I've learned that a square number can be multiplied by itself twice and be used as a base raised to the exponent of $2$, such as $3^2, 9^2, 0^2,$ etc. Square numbers can also fit into square shapes. ...
0
votes
1answer
1k views

How is it possible that two and three are the only consecutive prime numbers?

I've learned that two and three are not only two consecutive numbers, but are also two consecutive prime numbers. How is this possible? I think I'm on the right track in the following text: The ...
1
vote
1answer
44 views

Number Theory - Congruence implication proof

How do you prove that a congruence (such as this) implies another? I'm not sure where to begin and I think I'm missing something simple. $$ a \equiv b \ \text {mod} \ m \implies 4a \equiv 4b \ \text{...
0
votes
1answer
51 views

What book do you recommend for a better understanding of Euler Totient function?

Could anyone recommend a best(simple to read) book for Euler Totient function? If you have to know, it has been a while, since I introduced myself to this function. So please make sure that the book ...
2
votes
2answers
311 views

How to Solve Problem Similar to IMO(1995) Problem

Question: Let $ n$ be an postive integer number. How many $ n$-element subsets $A$ of $ \{1,2,\dots,2n\}$ are there such that $1+2+\cdots+2n$ is divisible by the sum of the elements of $A$. I ...
-1
votes
1answer
41 views

Proving a certain set is inductive?

Let $m$ be a natural number in a field $F$ and let $$ S_m= \{k:k\in N \mbox{ and } k\leq m \}\cup\{x:x\in F, m<x\} $$ Show that the set $S_m$ is inductive. Thanks in advance!
0
votes
1answer
99 views

Generate product of two primes that starts with 1234

A polynomial time algorithm to generate a composite number $N$ that starts with 1234 and has two prime factors. Each prime factor should have about $n$ digits with $n\approx 100$. To generalize, The ...
1
vote
0answers
43 views

Undecidebility in Number Theory [duplicate]

Recently one of my teachers says that it is not impossible that we find a problem in number theory that is undecidable in usual system of set theory. This was so wonderful for me. When I say this ...
2
votes
1answer
44 views

“Closeness” Definition writing for sets

I am having trouble writing the definition for this example: The closeness of the set {2,3,5} to 0 is 2. The closeness of the set {1.31,1.301,1.3001..} is 1.3. The closeness of the set (3,5] to 0 is ...
3
votes
1answer
98 views

What are the integer solutions of the system $a^2+b^2=c^2$, $a^3+b^3+c^3=d^3$?

How to solve these equations to find the integer numbers (a, b, c, and d)? $$a^2+b^2=c^2\tag{1}$$ $$a^3+b^3+c^3=d^3\tag{2}$$ I know one of solutions which is $a=3, b=4, c=5, d=...
0
votes
0answers
32 views

Is $\{(\frac{3}{2})^{n}\}_{\mathbb{N}}$ is equidistributed in [0,1] (related open question)

Is $\{(\frac{3}{2})^{n}\}$ for $n\in \mathbb{N}$ dense in [0,1] (open question). By $\{(\frac{3}{2})^{n}\}$ I mean the fractional part of $(\frac{3}{2})^{n}$. A more general question is: Is $\{(\frac{...
0
votes
1answer
90 views

Using the ABC-conjecture

I have to answer the following question: Let $a,b,c \in \mathrm{Z}_{\geq3}$, use the ABC conjecture to show (we suppose that the conjecture is true) that $x^ay^b-z^c=1$ has finite solutions for $x,y,...
-2
votes
0answers
30 views

A question about prime and factorization [duplicate]

Find a prime number $p$ and an integer $b < p$ such that $p|(b^{pāˆ’1} āˆ’ 1)$. Need help guys
1
vote
0answers
86 views

When $Ax^2+By^2=z^2$ has a solution in integers?

Consider the Diophantine equation $Ax^2+By^2=z^2$, with positive integer parameters $A$ and $B$ (not necessarily square-free or co-prime). When does this equation have a non-trivial solution? Can one ...
0
votes
2answers
51 views

Congruence mod396

A question in to the numbertheory: give a non-trivial soloution for the equation $x^6\equiv x\ mod\ 396$. And how many soloutions does this congruence have? I know by the Chinese Remainder Theorem ...
1
vote
1answer
24 views

Simple Modular Arithematic with Negative Numbers

If given an equation in the form: 3 = x mod 13 I know that I can generate a solution set by doing: X = 13q + 3 And ...
1
vote
0answers
112 views

What does $p\mathbb{Z}_p$ mean?

I am looking at Hensel's Lemma: Let $F(x)=a_0+a_1x+ \dots + a_n x^n \in \mathbb{Z}_p[x]$. We suppose that there is a $p$-adic number ($p>2$) $\alpha_1 \in \mathbb{Z}_p$, such that: $$F(\alpha_1) ...
3
votes
1answer
141 views

how to prove $\prod_{k=1}^{p-1} \sin(\frac{\pi k}{p}) = \frac{p}{2^{p-1}}$? [duplicate]

i found this relation whilst trying to evaluate the norm (over $\mathbb{Q}$) of $1-\zeta$ for $\zeta$ a primitive $p$-th root of unity ($p$ supposed prime) $$ \prod_{k=1}^{p-1} \sin(\frac{\pi k}{p}) = ...
0
votes
1answer
32 views

Find $\gcd (a,2^a)$ for all $a\in \mathbb{Z_+}$

We know that if $a$ is even then $(a,2^a)=a$ since $a\le 2^a $ for all $a\in \mathbb{Z} $ and if $a$ is odd then $(a,2^a)=1$ $\square$ Is this enough,and how would we work out $(...
0
votes
1answer
101 views

Proof of simple relation involving near primes?

Motivation (can skip!). (*) $\sum\log n \approx n\log n-n,$ and $$\sum\log n = \sum_{p_1\leq n} \log p_1+\sum_{p_2\leq n} \log p_2+...+\sum_{p_m\leq n} \log p_m$$ in which $p_k$ are numbers comprised ...
3
votes
2answers
104 views

Existence of integer solution to 63x+70y+15z=2010

I have an equation $63x+70y+15z=2010$. The question asks me to conclude whether it has an integral solution or not? Any help on how to proceed?
1
vote
1answer
94 views

convergence of the sequence $10^{-n}$ in the p-adic numbers

Let $p$ be prime. I am tasked to prove that the sequence $10^{-n}$ does not converge in $\mathbb{Q}_{p}$ for any $p$ where $\mathbb{Q}_{p}$ is the set of p-adic numbers. For $p=2$ or $5$, we see ...
1
vote
1answer
47 views

Equation $x^2 + y^2 + 1 = 0$ (mod $p$)

How to prove that equation $x^2 + y^2 + 1 = 0$ (mod $p$) has roots? Hints are acceptable.
4
votes
1answer
192 views

How to prove that there exist infinitely $(m,n)$ such $[n\sqrt{p}]=\frac{1}{2}(3m^2-m)$

Show that: for any prime number $p$, there are infinitely many pairs of positive integers $(m, n)$ such that $$[n\sqrt{p}]=\dfrac{3m^2-m}{2}$$ where $[x]$ is the largest integer not greater than $...
3
votes
1answer
192 views

Determinant value of a square matrix whose each entry is the g.c.d. of row and column position

Let $A=(a_{ij})$ be a $n \times n$ matrix with $a_{ij}=\gcd(i,j) , \forall i,j=1,2, \cdots, n$ , then how do we prove $\det A=\prod_{i=1}^n \phi(i)$ ? , where $\phi$ is the Euler's phi function
-2
votes
1answer
26 views

Polynomials over a finite field - avaerage value [closed]

Need to prove that if $$ \lim_{n\rightarrow\infty}\frac{1}{q^{n}}\sum_{\begin{array}[t]{c} f\textrm{ monic}\\ \deg(f)=n \end{array}}h(f) $$ exists, it's equal to: $$ \lim_{n\rightarrow\infty}\frac{...