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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38
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1answer
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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 ...
37
votes
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 ...
37
votes
3answers
1k views

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
1k views

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 ...
36
votes
2answers
4k views

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 ...
36
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8answers
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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 $n\geq 3$ ...
36
votes
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 ...
36
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1answer
718 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
votes
2answers
907 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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1answer
1k views

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
votes
2answers
967 views

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 ...
35
votes
1answer
1k views

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 ...
35
votes
1answer
514 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 : ...
35
votes
3answers
684 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
2k views

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
votes
2answers
1k views

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 ...
35
votes
1answer
834 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 ...
34
votes
2answers
558 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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1answer
2k views

$n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesn't use Chebyshev's theorem?

If $n\geq2$, then $n!$ is not a perfect square. The proof of this follows easily from Chebyshev's theorem, which states that for any positive integer $n$ there exists a prime strictly between $n$ and ...
33
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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
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1answer
738 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 ...
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0answers
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Is this similarity to the Fourier transform of the von Mangoldt function real?

Mathematica knows that the logarithm of $n$ is: $$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$ The von Mangoldt function should then be: ...
32
votes
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?
32
votes
5answers
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What is so interesting about the zeroes of the $\zeta$ 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 ...
32
votes
4answers
681 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 ...
32
votes
1answer
471 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
1answer
869 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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0answers
569 views

Gross-Zagier formulae outside of number theory

(Edit: I have asked this question on MO.) The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer ...
31
votes
9answers
4k views

How can you prove that the square root of two is irrational?

I have read a few proofs that $\sqrt{2}$ is irrational. I have never, however, been able to really grasp what they were talking about. Is there a simplified proof that $\sqrt{2}$ is irrational?
31
votes
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 ...
31
votes
1answer
828 views

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 ...
31
votes
2answers
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
2answers
481 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
3k 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 ...
31
votes
2answers
799 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
votes
3answers
2k views

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 ...
30
votes
3answers
495 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
3answers
478 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 ...
30
votes
2answers
666 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
1answer
629 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 $$ ...
29
votes
4answers
1k views

Fermat's Last Theorem and Kummer's Objection

In 1847 Lamé had announced that he had proven Fermat's Last Theorem. This "proof" was based on the unique factorization in $\mathbb{Z}[e^{2\pi i/p}]$. However, Kummer, proved that when $p=23$ we do ...
29
votes
5answers
9k 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 ...
29
votes
3answers
1k views

Can the sums of two sequences of reciprocals of consecutive integers be equal?

I'm primarily a programmer, so forgive me if I don't know the proper nomenclature or notation. Last night, an old teacher of mine told me about a question that had caused some noodle-scratching for ...
29
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1answer
964 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 ...
29
votes
2answers
2k views

Does the equation $x^4+y^4+1 = z^2$ have a non-trivial solution?

The background of this question is this: Fermat proved that the equation, $$x^4+y^4 = z^2$$ has no solution in the positive integers. If we consider the near-miss, $$x^4+y^4-1 = z^2$$ then this has ...
29
votes
5answers
860 views

Can a Mersenne number ever be a Carmichael number?

Can a Mersenne number ever be a Carmichael number? More specifically, can a composite number $m$ of the form $2^n-1$ ever pass the test: $a^{m-1} \equiv 1 \mod m$ for all intergers $a >1$ ...
28
votes
3answers
572 views

Sequence of numbers with prime factorization $pq^2$

I've been considering the sequence of natural numbers with prime factorization $pq^2$, $p\neq q$; it begins 12, 18, 20, 28, 44, 45, ... and is A054753 in OEIS. I have two questions: What is the ...
28
votes
2answers
864 views

Does $2x^2-1=y^5$ have a solution in integers, with $y>1$?

In my solution to this MSE problem, I noted that $2x^2-1=y^5$ is unlikely to have solutions in integers with $y>1$. Recently, I've tried to find a proof, without success. Following Thomas ...
28
votes
1answer
682 views

Evaluating the sum $\lim_{n\to \infty}\sqrt[2]{2+\sqrt[3]{2+\sqrt[4]{2+\cdots+\sqrt[n]{2}}}}$

The following nested radical $$\lim_{n\to \infty}\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}$$ is known to converge to 2. We can consider a similar nested radical where the degree of the radicals increases: ...
28
votes
1answer
2k 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. Here is my list: The Riemann Hypothesis: A Resource ...