All Questions
Tagged with elementary-proofs nt.number-theory
92 questions
24
votes
2
answers
3k
views
A Putnam problem with a twist
This question is motivated by one of the problem set from this year's Putnam Examination. That is,
Problem. Let $S_1, S_2, \dots, S_{2^n-1}$ be the nonempty subsets of $\{1,2,\dots,n\}$ in some ...
- 43.2k
12
votes
1
answer
390
views
A set of prime numbers
Consider a non-empty set $S$ of primes, with the property that, for every finite subset $S'\subset S$, all the primes dividing $\left(\prod_{k\in S'}k\right)+1$ are in $S$.
For instance, it can ...
- 1,096
2
votes
1
answer
411
views
Vandermonde determinant: modulo
There is a lot of fascination with the Vandermonde determinant and for many good reasons and purposes. My current quest is more of number-theoretic.
QUESTION. Let $p\equiv 3$ (mod $4$) be a prime. ...
- 43.2k
2
votes
1
answer
839
views
Primality test for generalized Fermat numbers
This question is successor of Primality test for specific class of generalized Fermat numbers .
Can you provide a proof or a counterexample for the claim given below?
Inspired by Lucas–Lehmer–Riesel ...
- 2,661
0
votes
1
answer
242
views
$2$-adic valuations and sum of divisor function
Consider the sum of $k^{th}$-power of divisors of $n$, denoted
$$\sigma_k(n)=\sum_{d\vert n}d^k.$$
Let $\nu_p(x)$ stand for the $p$-adic valuation of the integer $x$.
The following appears to be ...
- 43.2k
3
votes
1
answer
383
views
Primality test for specific class of $N=k \cdot b^n-1$
This question is successor of Compositeness test for specific class of $N=k \cdot b^n-1$ .
Can you provide a proof or a counterexample to the following claim :
Let $P_m(x)=2^{-m}\cdot \left(\left(...
- 2,661
11
votes
2
answers
911
views
Primality test for specific class of Proth numbers
Can you provide a proof or a counterexample for the following claim :
Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$
Let $N=k\cdot 2^n+1$ such ...
- 2,661
-3
votes
1
answer
381
views
Is it true that there are infinite palindromic primes that when squared give palindromic number? [closed]
Can you prove that there are infinite palindromic primes that when squared give a palindromic number?
2
votes
0
answers
306
views
Conjectured initial values of Inkeri's primality test for Fermat numbers
This is a repost of this question .
Can you provide a proof or a counterexample to the claim given below ?
First , we shall give a definition of the Inkeri's primality test for Fermat numbers :
...
- 2,661
1
vote
1
answer
252
views
Another question on strengthening the Sylvester-Schur Theorem
I've found an argument that if valid suggests that there is always a prime $p > n$ that divides ${{x+n} \choose n}$ when $x > \pi(n)$.
As a math amateur, I am always doubtful about my results ...
- 1,049
1
vote
0
answers
148
views
Combination of $k$-powers and divisibility [closed]
Let $n$ be a positive integer. Determine integers, $n+1\leq r\leq 3n+2$, such that for all integers, $a_1,a_2,\dots,a_m$, $b_1,b_2,\dots,b_m$, satisfying the equations
$$
a_1b_1^k+a_2b_2^k+\cdots+...
- 59
2
votes
0
answers
65
views
Can this function satisfy Song conditions?
Let $r_{s,k}(n)$ be the number of representation of a naturel number $l$ as a sum of $s$ positive $k$-th powers.
Joung Min Song introduced some conditions to study asymtotic behavior of some positive ...
- 1,257
2
votes
1
answer
241
views
sequence generated with polynomials
Below is an old olympiad problem, that turned out to be notoriously hard, that we couldn't solve it. If anyone has a solution for it, I'd be grateful.
Let $P(x)=x+1$, and $Q(x)=x^2+1$. We consider ...
- 59
66
votes
3
answers
6k
views
Chebyshev polynomials of the first kind and primality testing
Can you provide a proof or a counterexample for the claim given below ?
Inspired by Agrawal's conjecture in this paper and by Theorem 4 in this paper I have formulated the following claim :
Let $...
- 2,661
3
votes
1
answer
200
views
On decompositions of integers as a linear combination of $(1, 2, 3,\ldots)$
Edited: Given integer $N\geq 0$, let $$I(N):=\Bigl\{(n_k)_{k\geq 1}\in {\mathbb N}^\infty \,:\, n_k\geq 0, \sum_{k\geq 1}kn_k = N \Bigr\}$$ be the set of all decompositions of $N$ as a linear ...
- 251
5
votes
1
answer
275
views
An elementary question about a sequence of numbers
Let $\lambda_n$ be an increasing and unbounded sequence of positive real numbers and $a_n$ be a sequence of real numbers such that
$$\sum_{n=1}^\infty a_n \lambda_n^k=0 \ \ \text{ for all }\ \ k\geq ...
1
vote
1
answer
502
views
A Zsigmondy-theorem-analogy in the generalized Collatz-problem $3x+\rho$?
Remark : I've found a rather trivial answer for this question and so very likely the premise of paralleling it with the Zsigmondy-theorem is wrong, so this question might better be retracted. I'll ...
- 5,342
17
votes
0
answers
891
views
An elementary proof that, for every fixed $n \in \mathbf N^+$, there are infinitely many primes $\equiv -1 \bmod n$
This morning, I made a comment to a comment to a question of Ayman Moussa, only to point out that, among many others, there is an elementary proof of Dirichlet's theorem on the existence of infinitely ...
- 10.5k
7
votes
3
answers
2k
views
Solution to a Diophantine equation
Find all the non-trivial integer solutions to the equation
$$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=4.$$
- 403
2
votes
2
answers
707
views
Are simplified elementary proofs if valid interesting to the professional mathematical community [closed]
For the last 10+ years, as a math amateur, I worked nightly on understanding the distribution of primes and the classic results in the history of Fermat's Last Theorem.
I have made numerous mistakes ...
- 1,049
6
votes
1
answer
826
views
Going beyond the Sylvester and Schur theorem with regard to $x,x+1,\dots,x+n-1$
I was recent reading through Paul Erdos's classic elementary proof of Sylvester-Schur. It occurred me that there is a simple argument that when $x$ is sufficiently large and if $p_i$ represents the $...
- 1,049
16
votes
4
answers
2k
views
What can be said about this double sum?
Question. Can this number be expressed in terms of classical values?
$$\sum_{n,m=1}^{\infty}\frac1{(n^2+m^2)^{\frac32}}=1.056348517615643291\dots$$
UPDATE. I'm encouraged by Noam, Kevin and Igor's ...
- 43.2k
32
votes
3
answers
8k
views
Ideas in the elementary proof of the prime number theorem (Selberg / Erdős)
I'm reading the elementary proof of prime number theorem (Selberg / Erdős, around 1949).
One key step is to prove that, with $\vartheta(x) = \sum_{p\leq x} \log p$,
$$(1) \qquad\qquad \vartheta(x) \...
- 587
6
votes
1
answer
835
views
Beauty of some numbers discovered by Ramanujan
I am a graduate PhD student and my topic is analytic number theory. I am also a mathematics teacher. I am planning to give a course to pupils in high school that motivates them to study arithmetic and ...
- 1,257
1
vote
0
answers
118
views
Carry operations when adding two numbers [closed]
Let $x$ be a large positive real number and let $q\leq 2$ be a positive integer. It is known that for a positive integer $n,$ there exists a unique sequence $\left\{0\leq n_k\leq q-1\right\}_{k\geq 0}$...
- 1,257
1
vote
0
answers
104
views
Digits of sums of two integers [closed]
Let $q$ be a non-negative integer $\geq 2.$ For a non-negative integer $n$ It is known that there exixts a unique sequence of integer $0\leq n_k \leq q-1$ such that $$n=\sum_{k=0}^{+\infty} n_k q^k.$$
...
- 1,257
4
votes
1
answer
246
views
Independence of radicals: First-principles proof of special case
Reposting from MathStackexchange, original post is here, but got no answer.
I've known this problem for a long time:
Problem. Show that the number $\alpha=\sqrt{1} + \sqrt{2} + \ldots + \sqrt{n}$ is ...
- 997
1
vote
1
answer
199
views
Effective estimate for this infinite product over Hecke eigenvalues
Let $f$ be a primitive form of an even weight $k\geq 2$ for the full modular group $SL_2(Z)$ and let $\lambda_f(n)$ be the $n$-th normalized Fourier coefficient of $f.$ Can someone provide me with an ...
- 1,257
1
vote
2
answers
288
views
Implicit constant in Tenenbaum's result
In his famous book 'Introduction to Analytic and Probabilistic Number Theory', Gérald Tenenbaum established the following result (Theorem III.3.5):
Let $g$ be a positive multiplicative function and ...
- 1,257
0
votes
1
answer
374
views
Are there infinitely many $k$ for which $\frac{\sigma(k)}{k}=n^p$ and $p$ is an odd prime? [closed]
I would like to know if there are infinitely many $k$ for which $$\sigma(k)/k=n^p$$ such that $m=k{n}^{p-1}$ with $m,n>0$ and $p$ is an odd prime?
Note: $\sigma(\frac{m}{{n}^{p-1}})$ is the sum of ...
- 414
11
votes
1
answer
625
views
A congruence conjecture regarding $(r-s)^4-1 \equiv 0\!\pmod{4r^2s}$
Is the following conjecture true?
Conjecture. If $r > s \ge 1$ are relatively prime integers such that
\begin{equation}
(r-s)^4-1 \equiv 0\!\pmod{4r^2s}, \tag{1}
\end{equation}
then $r-s = 1$ ...
- 1,055
5
votes
0
answers
586
views
Primality test for specific class of generalized Fermat numbers
Can you provide a proof or a counterexample for the following claim :
Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ .
Let $F_{p,n}= (2p)^{2^n}+1 $ where $p$ is a prime number ...
- 2,661
3
votes
1
answer
292
views
Estimate sum with Euler function
(Note: this question was posted also in MSE)
I'd like to know if there's a closed formula or at least an estimate for the following (finite) sum:
$$
\sum_{D|p-1} \varphi(D) \,\varphi\left(\frac{p-1}{...
- 215
1
vote
0
answers
207
views
Proofs for almost prime limits
A number $n$ with prime factorization $$n=\prod_{i=1}^rp_i^{a_i}$$
is a k-almost prime if it has a sum of exponents $$\sum_{i=1}^{r}a_i=k$$ i.e., when the prime factor (multiprimality) function $\...
- 1,903
13
votes
3
answers
3k
views
Solving the quartic equation $r^4 + 4r^3s - 6r^2s^2 - 4rs^3 + s^4 = 1$
I'm working on solving the quartic Diophantine equation in the title. Calculations in maxima imply that the only integer solutions are
\begin{equation}
(r,s) \in \{(-3, -2), (-2, 3), (-1, 0), (0, -1),...
3
votes
1
answer
372
views
Is there an easy proof of this equation related to simultaneous Pell equations?
Working with the famous Baker-Davenport system of simultaneous Pell equations
\begin{align}
3x^2-2 &= y^2, &
8x^2-7 &= z^2, \qquad(\star)
\end{align}
I am left, after a series of ...
- 1,055
17
votes
2
answers
1k
views
Are [Wieferich] primes the only solutions to the equation $2^{k-1} \equiv 1 \pmod{k^2}$?
While studying a certain Diophantine equation in the squarefree integer $k \ge 2$, I believe I have proven the necessary restriction
$$2^{k-1} \equiv 1\!\!\pmod{k^2}. \qquad(\star)$$
Based on what ...
- 1,055
13
votes
3
answers
1k
views
At what point would an elementary generalization of Bertrand's Postulate be interesting?
I know that in 1952 Jitsuro Nagura was able to show that there is always a prime between $k$ and $\frac{6k}{5}$ for $k > 24$.
At what point would an improvement on Nagura's result be interesting? ...
- 1,049
10
votes
2
answers
1k
views
Length of Hirzebruch continued fractions
Suppose $a,b$ are two natural numbers relatively prime to $n$ and to each other. Assume $n\geq ab+1$. Suppose further that $\frac{a}{b}\equiv k \pmod{n}$ for some $k\in \lbrace 1,2,\dots, n-1\rbrace$ ...
- 85.6k
36
votes
1
answer
2k
views
On a remark of Tait on FLT for the exponent 3
This is one of those recreational questions that aren't really about research. I found a curious remark in an old volume of American Mathematical Monthly (1922) which I'll quote below:
In the ...
- 85.6k
16
votes
5
answers
9k
views
Elementary proof of the equidistribution theorem
I'm looking for references to (as many as possible) elementary proofs of the Weyl's equidistribution theorem, i.e., the statement that the sequence $\alpha, 2\alpha, 3\alpha, \ldots \mod 1$ is ...
- 169
27
votes
4
answers
11k
views
Is there an elementary way to find the integer solutions to $x^2-y^3=1$?
I gave this problem to my undergraduate assistant, as I saw that Euler had originally solved it (although I am having trouble finding his proof). After working on it for two weeks, we boiled the hard ...
- 18.7k