Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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5
votes
1answer
237 views

The Diophantine equation, $x^2 - y^2 = z^3$

I was recently studying some Diophantine equations and the equation $x^2 - y^2 = z^3$ caught my eye.I knew that $x=(n+1)(n+2)/2$ , $y=n(n+1)/2$ , $z=(n+1)$ where $n \in \mathbb{N}$ gives a set of ...
5
votes
5answers
938 views

Multiplicative inverse of a complex number.

Find the multiplicative inverse of $1+ 3\sqrt{2}$ in the ring $\mathbb{Q}(\sqrt{2})$ and use it to solve the equation $(1+3\sqrt{2})x=1-5\sqrt{2}$. I think that the inverse is the conjugate, so it ...
2
votes
1answer
173 views

Field Extension of the Rationals

So I'm considering a Field $\mathbb{F}$, such that $\mathbb{Q}$ is a subset of $\mathbb{F}$ and when it's considered a vector space over $\mathbb{Q}$, it has dimension 2. I want to show two things: ...
2
votes
3answers
313 views

Least Common Multiple of Fractions

how do i find lcm of two fractions? For example: $\frac{2}{3}$ and $\frac{5}{8}$
6
votes
5answers
709 views

How does one read aloud Vinogradov's notation $\ll$ and $\ll_{\epsilon }$?

How does one read aloud the Vinogradov's notation $\ll$ and $\ll_{\epsilon }$ as in $$f(x)\ll g(x)$$ and $$c\ll_{\epsilon }\left( \prod\limits_{p\mid abc}p\right) ^{1+\epsilon}.$$ Is the first one ...
3
votes
2answers
234 views

Find n that : $1+5u_nu_{n+1}=k^2, k \in N$

Let ${u_n}$ be such that: $$\begin{cases}u_1=20;\\u_2=30;\\ u_{n+2}=3u_{n+1}-u_{n},\; n \in \mathbb N^*.\end{cases}$$ Find $n$ such that: $$1+5u_nu_{n+1}=k^2,\; k \in \mathbb N.$$
0
votes
2answers
141 views

Use Lucas' test with $a=7$ and prove $71$ is prime

My working so far: $71-1=70$ and Prime factors of $70$ are $2 \times 5 \times 7$ Check $a=7$: $7^{(\frac{70}{2})} \equiv 7^{35} \equiv x (mod 71)$ How do I find $x$? Usually I would use Fermat's ...
9
votes
5answers
326 views

Determine the number of solutions of the equation $n^m = m^n$ [duplicate]

Possible Duplicate: $x^y = y^x$ for integers $x$ and $y$ Determine the number of solutions of the equation $n^m = m^n$ where both m and n are integers.
1
vote
1answer
66 views

How to get the result using Pepin's test

How do I achieve this result via Pepin's test by using Euler's Theorem and such to simplify $3^{2^{31}}$ and get the desired congruence of 10,324,303? $3^{(F_5-1)/2} = 3^{2^{31}} = 3^{2,146,483,648} ...
1
vote
4answers
268 views

Elementary Number Theory. Test Question.

If $a,b$ greater than or equal to 1 and if the $\text{gcd}(a,b)=1$ explain how we know that the $\text{lcm}(a,b)=ab$.
1
vote
3answers
121 views

Rational number solution for an equation

Does there exist $v=(a,b,c)\in\mathbb{Q^3}$ with none of $v$'s terms being zero s.t. $ a+b\sqrt[3]2+c\sqrt[3]4=0$ ? And I was doing undergraduate algebra 2 homework when I encountered it in my head. ...
1
vote
2answers
60 views

Polynomial factors

Why must $x^2 + x + 1$ be a factor of $x^5+x^4+x^3+x^2+x+1$? I know that when we divide $x^5+x^4+x^3+x^2+x+1$ by $x^3+1$ we get $x^2 + x + 1$, but is there an argument/theorem or anything that ...
1
vote
2answers
1k views

Find all integer solution of the equations:

Find all integer solutions of the equations: \begin{cases} 6x+21y&=&33 \\ 14x-49y&=&13 \end{cases} I'm not sure how to find all integer solutions for a, but I know there are no ...
0
votes
4answers
1k views

Prove that if c is a common divisor of a and b then c divides the gcd of a and b..

If $c$ is a common divisor of $a$ and $b$ then $c$ divides the greatest common divisor of $a$ and $b$. What can we use to prove this?
1
vote
3answers
455 views

Elementary number theory, if a divides b show that..

If $a$ divides $b$ argue that $a^3$ divides $b^5c$ for any $c$.
2
votes
2answers
404 views

How to find integer solutions for $x^3 - y^2 = 0 $?

How can I find integer solutions for $x^3 - y^2 = 0 $ ? In case that there are infinite number of solutions .How can we prove that ? and how to generate first few solutions ?
0
votes
1answer
327 views

How to find integer solutions for an ellipse equation?

How can I find the positive integer solutions to $x$ and $y$, given the integers $a$, $b$ and $c$ in the following ellipse equation in the form: $\frac{x^2}{a^2} + \frac{y^2}{b^2}=c$ For example, ...
0
votes
2answers
101 views

Elementary Number Theory.. If a divides..

If $a \mid (2c+3d)$ and if $-a \mid (c+d)$ then show that $3a \mid 3c$. The only progress I can say I've made is that the question is basically asking to show that $a \mid c$, because the $3$ is only ...
0
votes
1answer
110 views

Elementary Number Theory.. If a divides [duplicate]

Possible Duplicate: Elementary Number Theory.. If a divides.. If a divides (2c+3d) and if (-a) divides (c+d) then show that 3a divides 3c. The only progres I can say i've made is that it ...
2
votes
1answer
109 views

Minimal polynomial of the form $\zeta_p+\frac{1}{\zeta_p}+\zeta_q+\frac{1}{\zeta_q }$?

We can calculate the minimal polynomial of $ 2cos(\frac{2\pi}{7})=\zeta_7+\frac{1}{\zeta_7}$ over Q as x^3+x^-2x-1 and simlary for $2cos(\frac{2\pi}{5})=\zeta_5+\frac{1}{\zeta_5 }$. Now my question ...
1
vote
1answer
53 views

Inequality with numbers

It seems its a simple question, but I am confused. Let a be natural number and let b be some number $1\le b\le a$. Find an upper bound for $$ \frac{a^2+2b^2-4ab-a}{a(a-1)}. $$ I've got $$ ...
3
votes
3answers
382 views

Characterising reals with terminating decimal expansions

Show that a number has a terminating decimal expansion if and only if, it is rational and when in lowest terms, its denominator is coprime to all primes other than $2$ and $5$. This is an unsolved ...
6
votes
1answer
1k views

Difference between sum of even positioned digits and sum of odd positioned digits in a number is equal to 1

Numbers whose difference between Sum of digits at even location and Sum of digits at odd location is 1... Let us call those numbers that satisfy these condition to be "GOOD NUMBERS" For ex..) the ...
4
votes
1answer
1k views

How did Euclid prove Euclid's Lemma

In Elements, Book VII, Proposition 7, Euclid states: If a number is that part of a number which a subtracted number is of a subtracted number, then the remainder is also the same part of the remainder ...
4
votes
5answers
176 views

$p$ a prime, $p \equiv 3 \pmod 4$. Prove that $\frac{p-1}{2}! \equiv \pm 1 \pmod p$

Let $p$ be a prime, $p \equiv 3 \pmod 4$. Prove that $\frac{p-1}{2}! \equiv \pm 1 \pmod p$. This is an exercise in one of my lecture notes. I couldn't seem to figure out how to do it. We've just ...
4
votes
1answer
177 views

How to find odd solutions only for Pell's equation $x^2 - Dy^2 = 1$?

How can I find only the odd solutions for Pell's equation: $$x^2 - Dy^2 = 1$$ Specifically where $x$ is odd (but $y$ may be even or odd). Is there a way to generate the odd solutions to $x$, and can ...
0
votes
0answers
148 views

What is the sixth Martin quadruple?

Define a Martin quadruple {a,b,c,d} as a solution in non-zero integers to the system, $a+b+c+d = x^2$ $a^2+b^2+c^2+d^2 = y^2$ $a^3+b^3+c^3+d^3 = z^3$ It can be shown that there are an infinite ...
7
votes
1answer
185 views

the number field $\mathbb{Q}(\cos \frac \pi n)$

Let $x$ be $\cos \displaystyle \frac \pi n$ for some natural number $n$. Then is it true that $\mathbb{Q}(x^2+x)=\mathbb{Q}(x)$?
1
vote
1answer
93 views

Series of Mersenne primes

If the 'Lenstra - Pomerance - Wagstaff' conjecture is true, there are infinite Mersenne primes. In this case, if we consider the series: $$S_N=\sum_{k=1}^N \frac{1 }{M_k}$$ where $M_k$ is $k^{th}$ ...
1
vote
4answers
2k views

Get numbers that have only 2,3 and 5 as prime factors

I am given an integer N. I have to find first N elements that are divisible by 2,3 or 5, but not by any other prime number. ...
2
votes
1answer
108 views

Recapturing + on Natural numbers.

Consider the (multiplically written) free commutative monoid $M$ on a countably infinite set $\mathcal P$ of generators (it is isomorphic to $(\mathbb N,\cdot)$ with the primes as generators, ...
2
votes
2answers
352 views

Reduction of polynomials mod p

1-Let $f_1,f_2\in\mathbb{Z}[X]$ be two different irreducible monic polynomials. Is it true that for almost all primes $p$ (that is, for all but a finite number of primes), the polynomials $\bar{f}_1$ ...
2
votes
2answers
148 views

$2AB$ is a perfect square and $A+B$ is not a perfect square

If :$A=1!2!\cdots 1002!$, and $B=1004 ! 1005!\cdots2006!$, how to prove that: a) $2AB$ is a perfect square b) $A+B$ is not a perfect square
23
votes
1answer
712 views

Is $0.23571113171923293137\dots$ transcendental?

Is the following number transcendental? $$0.23571113171923293137\dots$$(Obtained by writing prime numbers consecutively from left to right, in the decimal expansion)
22
votes
4answers
1k views

Can someone explain the ABC conjecture to me?

I am an undergrad and I know that the conjecture may have been proven recently. But in reading about it, I am entirely confused as to what it means and why it is important. I was hoping some of you ...
0
votes
0answers
93 views

Meaning of having a rational $m$-torsion point

Suppose I have an elliptic curve $E/\mathbb{Q}$. What does it mean when one says $E$ has a rational $m$-torsion point over $\mathbb{Q}$? What does this mean for the torsion subgroup, ...
1
vote
0answers
67 views

the 2-rank of field

Let the field $K=\mathbb{Q}(\sqrt{p_1}, \sqrt{p_2 q}, i)$ where $p_1, p_2 \equiv 1 \mod{4}$ and $q \equiv 3 \mod{4}$, kronecker(2,$p_1$)=1 and kronecker(2,$p_2$)=kronecker($p_1$,$p_2$) = ...
2
votes
1answer
75 views

Modulo equation : $\frac{n^{k+1}-1}{n-1} \equiv a{\pmod p}$

Can we have directly answer for this question : $$\frac{n^{k+1}-1}{n-1} \equiv a{\pmod p}$$ (p is a prime and k,n is a fixed number) My question is : with fixed number n, k and p, can we know value ...
3
votes
1answer
254 views

Solvability of $x^q=2\mod p$

I've been discussing a problem recently Let $p, q$ be primes. If $x^q\equiv2\pmod p$ has no solution then $p\equiv1\pmod q.$ This is not a bi-equivalence (though it is "nearly" one): there are 811 ...
3
votes
2answers
92 views

$\mathbb{Q}$-Linear Combinations of Floor Functions taking values in $\{0,1\}$

The linear combinations $\lfloor x \rfloor - 2 \lfloor \frac{x}{2} \rfloor$ and $\lfloor x \rfloor - \lfloor \tfrac{x}{2} \rfloor - \lfloor \tfrac{x}{3} \rfloor - \lfloor \tfrac{x}{5} \rfloor + ...
15
votes
1answer
981 views

If the abc conjecture has been proven what implication does that have for elliptic curve cryptography?

I am not a mathematician, but I was wondering if the proposed proof of the abc conjecture (PDF) by Shinichi Mochizuki of Kyoto University would contain insights and mathematical tools that would lead ...
1
vote
3answers
412 views

Finite or infinite set?

Due to my not-so-advanced math skills, this question may take a few attempts to state clearly: Consider the unordered pair (2-tuple) partitions of n (e.g. with n=4, we have {{4,0},{3,1},{2,2}}). ...
1
vote
1answer
180 views

Devising an Algorithm to find out if a natural number n is the sum of 2 squares

The title says it all. On my homework I am tasked with creating an algorithm that determines whether or not a natural number n can be written as the sum of two squares. The only stipulation I am ...
3
votes
2answers
689 views

How to demonstrate that there is no all-prime generating polynomial with rational cofficents?

It seems like there is no polynomial with finite variables known, which could generate all prime numbers, by integer assignments. Is there a proof that such polynomial can not exist and does anyone ...
2
votes
4answers
195 views

Rounding non-integer values to integer values without causing an 'overflow'

The following problem comes from a piece of software I am currently working on, but since it's partially a mathematical problem, I'll ask the question here. In my application I have a fixed value ...
1
vote
1answer
376 views

System of 3 linear congruences

Find all solutions: $$\begin{cases} x\equiv 39 \mod(189) \\ x\equiv 25 \mod(539) \\ x\equiv 399 \mod(1089) \end{cases}$$ But $189=3^3\cdot7$, $539=11\cdot7^2$ and $1089=3^2\cdot 11^2$, so I can't ...
1
vote
2answers
268 views

Function with a Modular Inverse

For a combinatorics problem I have a function, $h(x)$ that is always divisible by five, but it is calculated in pieces, e.g. $h(1) = 43 + 7$. The final function that I need is $f(x) = (h(x) / 5) ...
5
votes
2answers
207 views

Easy highschool number theory equation

I have some problems with my homework. We have two integers $a$ and $b$ which satisfy the equation $a^b - b^a = 1008$. Show that $a$ and $b$ have the same remainder when divided by $1008$.
4
votes
2answers
528 views

Trends in the distribution of reordered digits of Pi (OEIS A096566)

First let me try to describe in more details below the approach of "reordering" digits of Pi, which is used in OEIS A096566 https://oeis.org/A096566 and what I have done analyzing it so far. I am ...
4
votes
4answers
427 views

Automorphic numbers

Problem. We say that the $n$-digit number $x$ is automorphic iff $x^2\equiv x \mod(10^n)$. Prove that if $x$ is $n$-digit automorphic number then $(3x^2-2x^3)\mod(10^{2n})$ is $2n$-digit ...