# Questions tagged [number-theory]

Questions on advanced topics - beyond those in typical introductory courses: higher degree algebraic number and function fields, Diophantine equations, geometry of numbers / lattices, quadratic forms, discontinuous groups and and automorphic forms, Diophantine approximation, transcendental numbers, elliptic curves and arithmetic algebraic geometry, exponential and character sums, Zeta and L-functions, multiplicative and additive number theory, etc.

40,792 questions
Filter by
Sorted by
Tagged with
109 views

### Sequence whose convergence towards 1 would be equivalent to the collatz conjecture being true, I would like feedback

I think I found a sequence whose convergence towards 1 would be equivalent to the collatz conjecture being true, we only need to study odd numbers since from an even number we can have an odd number ...
1k views

40 views

### Are there any known results for Dirichlet series for trigonometric functions?

I'm currently working on a problem that required that I come up with a Dirichlet series for cos(s), where s is any complex number. After trying a couple different things, I arrived at the following ...
33 views

### Existence of strictly increasing sequences such that $a_n(a_n+1) \mid b_n^2 + 1$.

Show that there exists two strictly increasing sequences $(a_n)$ and $(b_n)$ such that $a_n(a_n + 1) \mid b_n^2 + 1$. I have no idea how to solve this problem. I've tried a few different approaches, ...
384 views

### system of congruences has a solution if GCD = 1

I have the following problem. Let m, n be arbitrary non zero integers. Show that $x ≡ a \pmod m$ $x ≡ b\pmod n$ has a solution if $\gcd(m, n)$ divides both $a$ and $b$. Also interested in the ...
8k views

### modulo version of the quadratic formula and Euler's criterion

Use the modulo version of the quadratic formula and Euler's criterion to decide if the following has a solution or not. $2x^2+5x+8 \equiv 0\pmod{37}$ I'm not sure how I would use what was being ...
33 views

### Bound on a sum of divisor function

I am currently reading a math paper in which the author writes : $$\sum_{l\mid n}l\sigma_1(l)\ll n^2\sigma_{-1}(n)$$ Where the function $\sigma_k$ is defined by $\sigma_k(n)=\sum_{d\mid n}d^k$ and ...
1 vote
522 views

925 views

### Difference between a Reduced Residue Class and a Reduced Residue System

Currently studying Dirichlets proof of Primes in arithmetic progressions. Is there a difference between a Reduced Residue Class and a Reduced Residue System? Any subset $R$ of the integers is called a ...
38 views

64 views

### Evaluation of nontrivial zeros of $\zeta$ in explicit formulae

Im sure this question is completely trivial, but I just want to check my understanding: For the various explicit formulae in Analytic Number Theory involving sums over the nontrivial zeros of $\zeta$ ...
146 views

### Smallest prime number not divides $n$

For each integer $n>2$, we define $p(n)$ to be the smallest prime number that does not divide $n$. Prove that $$\lim_{n\to \infty}\frac{p(n)}{n}=0.$$ My argument is: We only need to prove \lim_{...
318 views

### Is there a perfect square (other than 9) all of whose digits are 7, 8, or 9?

Clearly, $3^2=9$ is a perfect square, all of whose digits are $7$, $8$, or $9$. Are there any other perfect squares with this property? This is an interesting question that does not seem to be ...
1 vote
118 views

I have to determine the number of elements of order $≤ 2$ in $(\mathbb{Z}/2^n\mathbb{Z})^×$, and use this to find the rank and the elementary divisors of $(\mathbb{Z}/2^n\mathbb{Z})^×$ I know that $(\... 0 votes 0 answers 31 views ### What is the meaning of logarithmic density in layman's terms? The way that I understand natural density is that it is an "intuitive way of describing the size of a set", whereas cardinality is an "absolute way of describing the size of a set".... 0 votes 3 answers 213 views ### Prove that there are infinitely many triples$(𝑎,𝑏,𝑐)$of integers such that$a^2+b^2+c^2=a^3+b^3+c^3$I have already made the observation that since the domain is real numbers,$a^2+b^2+c^2$the three terms is always positive but$a^3+b^3+c^3$since its to an odd power can have negative terms implying ... 30 votes 5 answers 5k views ### When the product of dice rolls yields a square Succinct Question: Suppose you roll a fair six-sided die$n$times. What is the probability that the product of the rolls is a square? Context: I used this as one question in a course for elementary ... 3 votes 2 answers 178 views ### Proof of$a^3 - b^3 = c^3 + d^3$, where$a,b,c,d$all rational? Reading Wikipedia article on Diophantus, it says in a book that survived that he makes reference to a lost book called Porisms and the theorem stated in the title: the difference between the cubes of ... 0 votes 0 answers 40 views ### What does "almost all in the sense of logarithmic density" mean in Tao's Collatz paper? Tao's paper: "Almost all orbits of the Collatz map attain almost bounded values" (via arXiv.org) 1st related question/discussion, I did not understand: Meaning of "almost all" in ... 0 votes 1 answer 2k views ### Condition For 4th degree polynomial equation having positive roots Consider the biquadratic polynomial equation$\rho_0y^4+\rho_1y^3+\rho_2y^2+\rho_3y+\rho_4y=0$, where$\rho_0,\rho_1,\rho_2, \rho_4$are positive and$\rho_3$is negative. So by Descartes' rule of ... -1 votes 1 answer 116 views ### Prove that no rational solution of equation$x^2-y^2=1002$exist. [closed] I have been able to prove in case of integers$x$and$y$. My approach in case of rational numbers is as follows, Let$x=\frac{a_1}{b_1}$and$y=\frac{a_2}{b_2}$with$\gcd(a_1,b_1)=1$and$\gcd(a_2,...
In answering a question here, I had to make use of the unique polynomial $P_N(x) = \sum_{j=0}^{N-1} a_j x^j \in \mathbb{C}[x]$ whose value at any primitive $N^{th}$ root of unity is $1$, and whose ...