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

learn more… | top users | synonyms (1)

0
votes
3answers
34 views

divisibility application in solving problems

How can I show that $a(2a^2+7)$ is divisible by $3$ for every integer a? I just simplified it as follows. $$ a(2a^2+7) = 3a^3 - a^3 + 9a - 2a = 3a(a^2-3)-a(a^2+2) $$ But I cannot show that $a(a^2+2)$ ...
0
votes
0answers
41 views

Which of the following sets are sub lattices of $\mathbb{Z}^2$?

Here are the first three sets: $\{(x, y) \in \mathbb Z^2 : x + y = 1\}$. $\{(x, y) \in Z^2 : x + y = 0\} = S^2$. $\{(x,y) \in Z^2 :2\mid x\} = S^3$ I found that the first one is not a subgroup ...
0
votes
1answer
62 views

What class am I most prepared for?

I've only taken up to calc 3, discrete, and linear algebra. Which course am I most prepared for? I'm going to be taking differential equations and advanced calc, but I want to take a 3rd class. I can ...
11
votes
1answer
212 views

Prove that if $n(a^2+b^2+c^2)=abc$ then $2\mid n$

Is it true that if $n\in\mathbb N$ and the diophantine equation $$n(a^2+b^2+c^2)=abc,\\(a,b)=(b,c)=(c,a)=1\tag1$$ has positive integer solutions $a,b,c$, then $2\mid n$? I can prove that ...
0
votes
2answers
42 views

Trouble finding remainder for this problem expression? [duplicate]

$$\left[\frac{2222^{5555}}{7} + \frac{5555^{2222}}{7}\right]$$ Please guide me through steps. Thanx..
1
vote
1answer
47 views

How to show $(1^2)(3^2)(5^2)…((p-2)^2)=(-1)^{(p+1)/2}$

I want to show the above problem using Wilson's theorem, which I know is $(p-1)!\equiv(-1)$ mod p. If I start with this I get $1\dot{}2\dot{}3\dot{}...\dot{}(p-1)\equiv(-1)$ mod p, but I don't know ...
0
votes
0answers
41 views

Does this congruence have any solution: $x^{2}\equiv5\pmod{227}$?

I don't know how to start. Do I have to find a primitive root and use its powers to find when it's equal to 5?
-5
votes
2answers
42 views

Number theory proof HELP [duplicate]

Prove the following... For all natural numbers n, gcd (n, n +1) = 1. I think this can be proved by contridiction. I'm just not sure how.
0
votes
1answer
30 views

Help with Proving the following

For all natural numbers $n$, $(n, n +1) = 1$ Let $k$ be a natural number. Then there exists a natural number $n$ (which will be much larger than $k$) such that no natural number less than $k$ and ...
1
vote
3answers
28 views

The least number which leaves remainders 2, 3, 4, 5 and 6 on dividing by 3, 4, 5, 6 and 7 is

Problem - The least number which leaves remainders 2, 3, 4, 5 and 6 on dividing by 3, 4, 5, 6 and 7 is? Solution - Here 3-2 = 1, 4-3 = 1, 5-4 = 1 and so on. So required number is (LCM of 3, 4, 5, 6, ...
4
votes
4answers
105 views

Show $\left( n!\right)^2 > n^n$.

If $n > 2$, show that $$\left(n!\right)^2 > n^n$$ Although the problem is pretty obvious, I couldn't come up with a rigorous proof. I was thinking some sort of AM-GM, but couldn't build ...
1
vote
0answers
65 views

How to prove $\pi ^{3}$ is not constructible from the fact that $\pi $ is not constructible?

I know how to do this for $\sqrt[3]{\pi }$: First suppose it is constructible and then you just set it equal to $x_{0}=\sqrt[3]{\pi }$ and take the third power of both sides. Then you get ...
1
vote
2answers
39 views

How many solutions are there for $x^3 \equiv -1\mod(365)$?

For $mod(5)$, I found that there is only solution but for $mod(73)$, I'm a little confused. Factoring out $x^3 + 1=(x+1)(x^2-x+1)$, the second equation has no real solutions, hence $x\equiv-1$ is the ...
6
votes
1answer
106 views

Find all natural numbers such that $\sum_{k=1}^{n} \frac{n^k}{k!}$ is an integer

Find all natural numbers such that $\sum_{k=1}^{n} \frac{n^k}{k!}$ is an integer. I've tried to bring all fractions under commmon denominator and it didn't helped me much. With guessing I find out ...
1
vote
1answer
28 views

Is $n=2$ the only root of $M(n!)$…?

Wolfram can help till $n=9$, but are there other value larger than $2$ for which $$ M(n!)=0, $$ where $M(n)$ is Merten's function.
0
votes
2answers
33 views

Chinese remainder theorem when mod divisible with 2

I can't understand how chinese remainder theorem works when the all 3 mod numbers are divisible with 2. For example: x = 2 mod 6 x = 6 mod 10 x = 8 mod 12 * = ...
2
votes
1answer
39 views

consecutive prime power

I'm interesting on consecutive prime power numbers. I see that there is the Mersenne primes and the Fermat Primes that give solutions and $(8,9)$. In Sloane collection it is referred on A006549 and it ...
1
vote
2answers
67 views

Proof that $y^2=x^3+x$ has a unique integer solution

Prove that the equation $y^2=x^3+x$ has only one integer solution, namely $x=y=0$.
1
vote
1answer
79 views

Fermat last theorem

Can someone please give the original paper by Prof. Wiles of the proof of Fermat's Last Theorem? I cannot find it.
1
vote
1answer
18 views

On the number of midpoint free subsets

A set $X$ of real numbers is called midpoint free if whenever $x,y$ are distinct elements of $X$ then $\frac{x+y}2 \not \in X$. What is number of midpoint free subsets of $\{1,2,...,n\}$?
1
vote
1answer
39 views

If p $\equiv$ 3 (mod 4) with p prime, prove -1 is a non-quadratic residue modulo p.

If p $\equiv$ 3 (mod 4) with p prime, prove -1 is a non-quadratic residue modulo p. I suppose this would not be true if p $\equiv$ 1 (modulo 4). To prove something is a non-square I find to be ...
0
votes
1answer
46 views

Calculate the remainder of 5 to the power of 120

I was going through this thread And the first answer made me think. Fermat's Little Theorem tells us that $5^{18} = 1$ mod $19$. Observe next that $5^{120} = (5^{18})^6 \cdot 5^{12}$. ...
2
votes
2answers
316 views

In Fermat's little theorem, if mod is not prime?

Today I learned about Fermat's little theorem It says - Fermat's little theorem states that if p is a prime number, then for any integer a, the number a p − a is an integer multiple of p. In ...
0
votes
4answers
58 views

What is the remainder when 4 to the power 1000 is divided by 7

What is the remainder when $4^{1000}$ is divided by 7? In my book the problem is solved, but I am unable to understand the approach. Please help me understand - Solution - To find the ...
0
votes
2answers
67 views

Solve and explain diophantine equation

A Diophantine equation ax+by = c always has a solution whenever a and b are relatively prime. Find x ,y such that $$93x-81y=3 $$
0
votes
3answers
45 views

Method of solving extended Euclidean algorithm for three numbers?

I already got idea of solving gcd with three numbers. But I am wondering how to solve the extended Euclidean algorithm with three, such as: 47x + 64y + 70z = 1 ...
2
votes
2answers
57 views

Why is the Legendre symbol not defined for $p = 2$ (even prime)?

Why is the Legendre symbol not defined for $p = 2$ (even prime) ? According to the definition of the Legendre symbol $$\left(\frac a p\right)$$ it is defined for an odd prime $p$ only. Even thus ...
1
vote
1answer
26 views

Confused about discrete logarithm question

For purposes of explaining the notation for those unfamiliar, if we fix a prime $q$, as well as $a,b$ nonzero integers $\mod{q}$, $L_a(b) = x$ is the solution to the equation $b = a^x \mod{p}$ We are ...
1
vote
0answers
60 views

Proposition 17, p. 68 of Lang's Algebraic Number Theory

I'm stuck on a detail of the following propositon: Let $K, E$ be linearly disjoint number fields with degrees $n$ and $m$ over $\mathbb{Q}$ whose discriminants (over $\mathbb{Q}$) are relatively ...
1
vote
0answers
23 views

Finding a particular integral basis of the cyclotomic field

Let $\zeta_{39}$ be a primitive $39$th root of unity. How can I prove that all the conjugates of $\zeta_{39}$ form an integral basis of $\mathbb{Q}(\zeta_{39})$? This is from the paper "Cyclotomic ...
2
votes
0answers
33 views

Alternative Proof for “Roots of Mertens Function-Farey Sequence-Cosines Relations”

You can write Merten's function as $$ M(n)= \sum_{a\in \mathcal{F}_n} e^{2\pi i a} , $$ where $\mathcal{F}_n$ is the Farey sequence of order $n$. The sum may be split into imaginary and real ...
0
votes
0answers
25 views

Arithmetic functions in diophantine number theory

I am looking for exercises, theorems and proofs in diophantine number theory which make "good" (so not like just giving the number of units modulo n with $\varphi(n)$, but something more deeper) use ...
1
vote
1answer
47 views

A constant whose decimal digits are the terms of its infinite series? [closed]

Tell the constant whose decimal digits are the terms of its infinite series? Trott constant is such constant but that is for continued fraction. I want the constant for infinite sum. @putonhold staff ...
3
votes
1answer
47 views

Algorithm to find representation of an element of field extension $\mathbb{Q}(q)$ in the form $\sum a_i q^i$

Let $\mathbb{Q}(q)$ be a field extension of $\mathbb{Q}$, where $q$ is a real root of some monic irreducible polynomial $p(x) \in \mathbb{Z}[x]$ of degree $d=3$. Given $x \in \mathbb{R}$, (or ...
0
votes
4answers
42 views

Gcd and divisibility properties

suppose $a|c$ , $b|c$ and $gcd(a,b)=1$ .Then show that $ab|c$. Here $a,b,c$ are all real numbers.Can i start from the properties of divisibility as if $a|c$ and $c|b$ then $a|b$?
1
vote
0answers
53 views

Is 91 the only number which is both a centred cube number and a centred hexagon number?

The proof that $6^3 - 5^3 = 3^3 + 4^3$ is the only such identity involving consecutive numbers is elementary. I'm interested to know if a stronger result is true: whether 91 is the only number that is ...
0
votes
2answers
26 views

GCD of two real numbers

How would I show that gcd($2a+1 , 9a+4)=1 $? Here $a$ is an integer. I used the definition of the greatest common divisor, but felt it is too lengthy.
1
vote
2answers
51 views

How to show that if $\phi(n)$ equals to n itself, then n must be 1?

That is: If $\phi(n) = n$ then $n = 1$ Could someone give me a clue?
0
votes
1answer
41 views

Rational Approximation

Suppose I have a very large integer $N$ and $a_1,....a_k$ are integers $(a_i,N)=1$ for any $1\le i\le k.$ Suppose also that $k$ is small compared to $N$ (as small as we wish). Does there exist an ...
0
votes
1answer
48 views

functions determined by charachters are linearly independent?

Let $X$ be a set with an action of $\mathbb{Z}/N\mathbb{Z}$. For a Dirichlet character $\chi$ mod N we set $$R(\chi)=\left\{ f:X \to \mathbb{C} ~\mid~ f(l s)=\chi(l)f(s) \text{ for all } l\in ...
0
votes
0answers
33 views

Constant in an inequality for a height of an elliptic curve

I am trying to find explicitly a constant $\kappa$ in an inequality for the height of an elliptic curve. Suppose the curve $E$ is defined by $y^2 = x^3 - kx$ with $k \neq 0$, the curve is defined over ...
2
votes
1answer
40 views

Derivation of the lower bound of Euler's Phi Function

In the answer to a different question titled "Is the Euler phi function bounded below?", one answer derives the fact that if $0<\delta<1$, then $\frac{\phi(n)}{n^{1-\delta}}$ attains its minimum ...
1
vote
3answers
47 views

On special integer gaps.

Calling an integer square-in if it is not square-free or a square. If $A$ and $B$ are two consecutive odd square-in integers , $A\gt B$ , 9 does not divide B . Can $A-B\lt 12$ ?
0
votes
0answers
36 views

Solutions to Diophantine equations

I have been thinking about Pell’s equation and, in particular, the case $x^2-61y^2=1$ (eq. 1) The exponents (2) and the factor 61 seem intuitively out of proportion to the very high values for the ...
1
vote
3answers
74 views

Find the smallest k.

Find the smallest $k$, $$\sum_{n = 0}^{1013} \binom{2n}{n}k^n \mod{2027} \equiv 0$$ The problem was posted on Brilliant, but no one has submitted a solution yet. I tried expanding to get an idea, ...
0
votes
2answers
30 views

sequence $a_n = \lceil \sqrt{2}n \rceil $

I was trying to prove $\lceil \sqrt{2}n \rceil + \lceil \sqrt{2}m \rceil \geq \lceil \sqrt{2}(n+m) \rceil$ where $m,n\in \mathbb{z}$ Direct proof I tried but could not figure out. I tried fixing m ...
1
vote
1answer
42 views

Convert a Number to a Sentence [closed]

My task is to take any number, and have it replaced with a sentence/string representing its value. To be exact, I want something along the lines of; 150 = hi 151 = hj 152 = hk This of course ...
1
vote
3answers
309 views

Find the smallest positive integer x such that 2015! ≡ x (mod 2017)

Q. The next year that is a prime is 2017. Find the smallest positive integer x such that 2015! ≡ x (mod 2017). So, this is what I have; By Wilson’s theorem, (2017-1)! ≡ -1 (mod 2017) ⇒ 2016! ≡ -1 ...
0
votes
1answer
23 views

show divisibility facts in detail number theory

Show that if $a | b$, $a > 0$ and $b > 0$, then $a\le b$ if $ac | bc$, then $a | b$ $∀ n ∈ \mathbb Z\colon 2 |(n^2-n) $ A Diophantine equation $$ax+by = c $$ always has a solution whenever ...
0
votes
0answers
38 views

The solutions to $x^2+y^2=5$ in $\mathbb{Q}$. [duplicate]

Consider the following equation: $$x^2+y^2=5.\tag{1}$$ What are the solutions to this equation if $x,y\in\mathbb{Q}$, where $\mathbb{Q}$ is the set of all rational numbers? My attempt: Because ...