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.

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Why is $\varphi$ called “the most irrational number”?

I have heard $\varphi$ called the most irrational number. Numbers are either irrational or not though, one cannot be more "irrational" in the sense of a number that can not be represented as a ratio ...
41
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5answers
2k views

Intuition for the Importance of Modular Forms

I am learning about modular forms for the first time this term and am just starting to wrap my head around what might be the big picture of things. I was wondering if the following interpretation of ...
41
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1answer
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Sums of prime powers

You are given positive integers N, m, and k. Is there a way to check if $$\sum_{\stackrel{p\le N}{p\text{ prime}}}p^k\equiv0\pmod m$$ faster than computing the (modular) sum? For concreteness, you ...
41
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1answer
1k views

Estimate for $n$th prime

A good approximation I have found for $p_{n}$ is \begin{align} \int_{2}^{n}\log (x \log (x \log (x)))\ dx\\ \end{align} and seems to be a better estimate than $n \log (n)$. The error term seems to ...
41
votes
1answer
865 views

Are there infinite many $n\in\mathbb N$ such that $\pi(n)=\sum_{p\leq\sqrt n}p$?

Are there infinite many $n\in\mathbb N$ such that $$\pi(n)=\sum_{p\leq\sqrt n}p,\tag{1}$$ where $\pi(n)$ is the Prime-counting_function? For example, ...
40
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6answers
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Is $\sqrt[3]{p+q\sqrt{3}}+\sqrt[3]{p-q\sqrt{3}}=n$, $(p,q,n)\in\mathbb{N} ^3$ solvable?

In this recent answer to this question by Eesu, Vladimir Reshetnikov proved that $$ \begin{equation} \left( 26+15\sqrt{3}\right) ^{1/3}+\left( 26-15\sqrt{3}\right) ^{1/3}=4.\tag{1} \end{equation} $$ ...
40
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1answer
846 views

A new continued fraction for Apéry's constant, $\zeta(3)$?

As a background, Ramanujan also gave a continued fraction for $\zeta(3)$ as $\zeta(3) = 1+\cfrac{1}{u_1+\cfrac{1^3}{1+\cfrac{1^3}{u_2+\cfrac{2^3}{1+\cfrac{2^3}{u_3 + \ddots}}}}}\tag{1}$ where the ...
40
votes
1answer
386 views

How many non-rational complex numbers $x$ have the property that $x^n$ and $(x+1)^n$ are rational?

In this answer, I proved the following statement: For each positive integer $n$, there are finitely many non-rational complex numbers $x$ such that $x^n$ and $(x+1)^n$ are rational. These complex ...
39
votes
2answers
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Why is it hard to prove whether $\pi+e$ is an irrational number?

From this list I came to know that it is hard to conclude $\pi+e$ is an irrational? Can somebody discuss with reference "Why this is hard ?" Is it still an open problem ? If yes it will be helpful ...
39
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5answers
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Generalizing the sum of consecutive cubes $\sum_{k=1}^n k^3 = \Big(\sum_{k=1}^n k\Big)^2$ to other odd powers

We have, $$\sum_{k=1}^n k^3 = \Big(\sum_{k=1}^n k\Big)^2$$ $$2\sum_{k=1}^n k^5 = -\Big(\sum_{k=1}^n k\Big)^2+3\Big(\sum_{k=1}^n k^2\Big)^2$$ $$2\sum_{k=1}^n k^7 = \Big(\sum_{k=1}^n ...
39
votes
3answers
698 views

Prove that there exist infinitely many integers $(n^{2015}+1)\mid n!$

I conjecture that there exist infinitely many integers $n$ such that $$(n^{2015}+1)\mid n!.$$ I have seen a simpler problem that there exist infinitely many integers $n$ such that $(n^2+1)\mid ...
39
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4answers
770 views

Partitioning the integers $1$ through $n$ so that the product of the elements in one set is equal to the sum of the elements in the other

It is known that, for $n \geq 5$, it is possible to partition the integers $\{1, 2, \ldots, n\}$ into two disjoint subsets such that the product of the elements in one set equals the sum of the ...
37
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9answers
1k views

Find all positive integers $n$ s.t. $3^n + 5^n$ is divisible by $n^2 - 1$

as is the question in the title, I am wishing to find all positive integers $n$ s.t. $3^n + 5^n$ is divisible by $n^2 - 1$. I have so far shown both expressions are divisible by $8$ for odd $n\geq 3$ ...
37
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1answer
637 views

Numbers $n$ such that the digit sums of $n, n^2,\cdots,n^k$ coincide.

Let $S(n)$ be the digit sum of $n\in\mathbb N$ in the decimal system. When I was playing with numbers, I noticed the followings : ...
37
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3answers
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Triangular Factorials

I came across a statement online and have been looking for a proof : It states that 1, 6 and 120 are the only numbers which are both triangular and factorials. Is there any way I can prove this? ...
37
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4answers
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How to prove $k!+(2k)!+\cdots+(nk)!$ has a prime divisor greater than $k!$

Question: Let $k$ be a positive integer. Show that there exist $n$ such that $$I=k!+(2k)!+(3k)!+\cdots+(nk)!$$ has a prime divisor $P$ such that $P>k!$. My idea: Let us denote by ...
37
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2answers
1k views

Prime powers, patterns similar to $\lbrace 0,1,0,2,0,1,0,3\ldots \rbrace$ and formulas for $\sigma_k(n)$

Some time ago when decomponsing the natural numbers, $\mathbb{N}$, in prime powes I noticed a pattern in their powers. Taking, for example, the numbers $\lbrace 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 ...
37
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3answers
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Small primes attract large primes

$$ \begin{align} 1100 & = 2\times2\times5\times5\times11 \\ 1101 & =3\times 367 \\ 1102 & =2\times19\times29 \\ 1103 & =1103 \\ 1104 & = 2\times2\times2\times2\times ...
37
votes
2answers
998 views

Proof that $123456789098765432111$ is prime?

The mathematician Charles Weibel asks on his home page the following "fun question": How can you prove that 123456789098765432111 is a prime number? (He notes the fact $$12345678987654321 = ...
36
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5answers
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What is so interesting about the zeroes of the Riemann $\zeta$ function?

The Riemann $\zeta$ function plays a significant role in number theory and is defined by $$\zeta(s) = \sum_{n=0}^\infty \frac{1}{n^s} \qquad \text{ for } s > 1 \text{ and } s= \sigma + it$$ The ...
36
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1answer
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Why does this test for Fibonacci work?

In order to test if a number $A$ is Fibonacci, all we need to do is compute $5A^2 + 4$ and $5A^2 -4$. If either of them is a perfect square, the number is Fibonacci, otherwise not. Why does this test ...
36
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1answer
783 views

Geometric intuition behind The Mordell Conjecture

The Mordell Conjecture/Faltings Theorem says roughly that if $K$ is an algebraic number field and $X$ is an algebraic curve defined over $K$ of genus $g >1$ then the set of $K$-rational points ...
36
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1answer
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Generalizing Ramanujan's proof of Bertrand's Postulate: Can Ramanujan's approach be used to show a prime between $4x$ and $5x$ for $x \ge 3$

Perhaps, I've been thinking too long about Ramanujan's proof, but it appears to me that his argument can be generalized beyond $x$ and $2x$. My argument below attempts to show that for $x \ge 1331$, ...
35
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1answer
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Is $\sqrt1+\sqrt2+\dots+\sqrt n$ ever an integer?

Related: Can a sum of square roots be an integer? Except for the obvious cases $n=0,1$, are there any values of $n$ such that $\sum_{k=1}^n\sqrt k$ is an integer? How does one even approach such ...
35
votes
3answers
771 views

The resemblance between Mordell's theorem and Dirichlet's unit theorem

The first one states that if $E/\mathbf Q$ is an elliptic curve, then $E(\mathbf Q)$ is a finitely generated abelian group. If $K/\mathbf Q$ is a number field, Dirichlet's theorem says (among other ...
35
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1answer
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What is the sum of sum of digits of $4444^{4444^{4444}}$?

A recent question asked about the sum of sum of sum of digits of $4444^{4444}$. The solution there works mainly because the number chosen is small enough for the sum of sum of sum to be equal to the ...
35
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2answers
593 views

Is $n(n+1)$ ever a factorial?

Brocard's problem asks if $(n-1)(n+1)$ is ever a factorial. My question is similar: is $n(n+1)$ ever a factorial? This can be seen as the special case $k=2$ of the question: for $2\le k\le n-2,$ when ...
34
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4answers
5k views

Would a proof to the Riemann Hypothesis affect security?

If a solution was found to the Riemann Hypothesis, would it have any effect on the security of things such as RSA protection? Would it make cracking large numbers easier?
34
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2answers
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Do all natural numbers have a nonzero multiple that is a palindrome in base 10?

Some natural numbers have a nonzero multiple that is a palindrome in base 10. For example, $106 \times 2 = 212$, which is a palindrome, and $29 \times 8 = 232$, which is also a palindrome. Aside ...
34
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4answers
752 views

$\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor$ is true?

I found the following relational expression by using computer: For any natural number $n$, $$\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor.$$ Note ...
34
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2answers
3k views

Books about the Riemann Hypothesis

I hope this question is appropriate for this forum. I am compiling a list of all books about the Riemann Hypothesis and Riemann's Zeta Function. The following are exluded: Books by mathematical ...
34
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1answer
602 views

Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

I've known that one can arrange all the numbers from $1$ to $\color{red}{15}$ in a row such that the sum of every two adjacent numbers is a perfect square. $$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$ ...
33
votes
5answers
11k views

Checking if a number is a Fibonacci or not?

The standard way (other than generating up to $N$) is to check if $(5N^2 + 4)$ or $(5N^2 - 4)$ is a perfect square. What is the mathematical logic behind this? Also, is there any other way for ...
33
votes
4answers
3k views

Does there exist rational $a,b,c$, such that $\sqrt[3]{1}+\sqrt[3]{2}+\sqrt[3]{4}=\sqrt[3]{a}+\sqrt[3]{b}+\sqrt[3]{c}$

Let $w = \sqrt[3]{1}+\sqrt[3]{2}+\sqrt[3]{4}$. How to prove that there are no triples $(a,b,c)$, such that $a,b,c \in \mathbb{Q}$; $a \leqslant b \leqslant c$; $(a,b,c)\ne (1,2,4)$; $w = ...
33
votes
1answer
817 views

How does one see Hecke Operators as helping to generalize Quadratic Reciprocity?

My question is really about how to think of Hecke operators as helping to generalize quadratic reciprocity. Quadratic reciprocity can be stated like this: Let $\rho: Gal(\mathbb{Q})\rightarrow ...
33
votes
2answers
5k views

Alternative proof that $(a^2+b^2)/(ab+1)$ is a square when it's an integer

Let $a,b$ be positive integers. When $$k = \frac{a^2 + b^2}{ab+1}$$ is an integer, it is a square. Proof 1: (Ngô Bảo Châu): Rearrange to get $a^2-akb+b^2-k=0$, as a quadratic in $a$ this has two ...
33
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1answer
955 views

Is there a power of 2 that, written backward, is a power of 5?

In this note the famous mathematical physicists Freeman Dyson gives an example of a true statement that is impossible to prove. Or so he states. The statement is as follow: Numbers that are exact ...
32
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7answers
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Uses of quadratic reciprocity theorem

I want to motivate the quadratic reciprocity theorem, which at first glance does not look too important to justify it being one of Gauss' favorites. So far I can think of two uses that are basic ...
32
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3answers
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How far are the $p$-adic numbers from being algebraically closed?

A few days ago I was recalling some facts about the $p$-adic numbers, for example the fact that the $p$-adic metric is an ultrametric implies very strongly that there is no order on $\mathbb{Q}_p$, as ...
32
votes
1answer
490 views

Is this sequence unbounded?

Problem. Suppose that a sequence $\{a_n\}_{(n\ge 1)}$ is a strict increasing sequence of positive integers such that $$\forall i,\phantom{;}j(i\neq j);\phantom{;}a_i \not\mid a_j$$ Prove that ...
32
votes
2answers
508 views

Are there/Why aren't there any simple groups with orders like this?

The orders of the simple groups (ignoring the matrix groups for which the problem is solved) all seem to be a lot like this: ...
31
votes
3answers
7k views

Yitang Zhang: Prime Gaps

Has anybody read Yitang Zhang's paper on prime gaps? Wired reports "$70$ million" at most, but I was wondering if the number was actually more specific. *EDIT*$^1$: Are there any experts here who ...
31
votes
3answers
530 views

Do we really know the reliability of PrimeQ[n] (for $n>10^{16}$)?

The algorithm Mathematica uses for its PrimeQ function is described on MathWorld. That web page says PrimeQ uses, "the multiple ...
31
votes
2answers
858 views

Can every number be written as a small sum of sums of squares?

In a practice for a programming competition, one problem asked us to find the smallest number of pyramids which can be built using exactly $n$ blocks, where pyramids have either $k\times k, ...
30
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1answer
1k views

How does $ \sum_{p<x} p^{-s} $ grow asymptotically for $ \text{Re}(s) < 1 $?

Note the $ p < x $ in the sum stands for all primes less than $ x $. I know that for $ s=1 $, $$ \sum_{p<x} \frac{1}{p} \sim \ln \ln x , $$ and for $ \mathrm{Re}(s) > 1 $, the partial sums ...
30
votes
3answers
511 views

Constructing $\mathbb N$ from the set of factorials

Let S be the set $\{0!, 1!, 2!, \ldots\}$. Is it possible to construct any positive integer using only addition, subtraction and multiplication, and using any element in S at most once? For example: ...
30
votes
2answers
744 views

Proof of no prime-representing polynomial in 2 variables

In "The New Book of Prime Number Records", Ribenboim reviews the known results on the degree and number of variables of prime-representing polynomials (those are polynomials such that the set of ...
30
votes
6answers
638 views

Other interesting consequences of $d=163$?

Question: Any other interesting consequences of $d=163$ having class number $h(-d)=1$ aside from the list below? Let $\tau = \tfrac{1+\sqrt{-163}}{2}$. We have (see notes at end of list), ...
30
votes
4answers
647 views

What is the *middle* digit of $3^{100000}$?

The decimal representation of $3^{100000}$ has $47713$ digits. What is the $23857^{th}$ digit - i.e. the one in the $10^{23856}$'s place? There are lots of questions on this site asking for the ...
30
votes
1answer
666 views

Very tight prime bounds

Is it possible that $$\left|\operatorname{li}(n)-\sum_{k=1}^{\lfloor\log(n)\rfloor}\dfrac{\pi(n^{1/k})}{k}-\log(2)-\dfrac{1}{2}\right|<\dfrac{2\sqrt{n}}{e\log(n)}?$$ Since $$ ...