# Tagged Questions

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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### Proof verification - $\sqrt[3]{2}$ is irrational

This might be a duplicate, but I have not found one on this site. This is my proof and I was wondering if this proof depends on it's conclusion. Here it is: Assume $\sqrt[3]{2}$ is rational. Then ...
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### coefficients that make $p(x)=(a_1x+b_1)^3+(a_2x+b_2)^3+(a_3x+b_3)^3+(a_4x+b_4)^3-x$ a constant

Find integers $a_i$ and $b_i$, $i=1,2,3,4$, such that $p(x)$ is a constant function: $p(x)=(a_1x+b_1)^3+(a_2x+b_2)^3+(a_3x+b_3)^3+(a_4x+b_4)^3-x$ I don't even know if such coefficients exist or not. ...
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### calculating differences between entries in a table using each entry or skipping entries after rounding

Entry two of a table is entry one plus or minus some number. Each entry is determined by adding or subtracting some value to/from the previous entry. The value varies. The entries are composed of ...
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### A trick for calculating $n^6$ that I don't understand

I was doing a math exercise and it asked to find what are the possible units digits of $n^6$ knowing that $n\in\mathbb Z$. The solution said that because we are concerned only with finding what the ...
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### Fermat's Last Theorem - Variation with arithmetically descending exponents

Are there solution(s) to the following variant of Fermat's Last Theorem in the positive integers? $$a^n + b^{n-i} = c^{n-2i}$$ I haven't been able to identify any trivial solutions. To my ...
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### Proof that if $\gcd(m,n) = 1$, then $\gcd(m+n,mn ) = 1$. [duplicate]

I need help with this excercise. If $\gcd(m,n) = 1$, then $\gcd(m+n,mn ) = 1$. I don't know how to prove this, I know the definition of $\gcd$ but I can't prove it.
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If you plot the following function $$f(n) = \left|\frac{(p_1+\ldots+p_{n})}{n} - \frac{(p_1+\ldots+p_{n-1})}{n-1}\right|$$ you get a graph that is similar to $$f(x) = \frac{5}{4}\log(x) + ... 0answers 22 views ### Implementing FizzBuzz game I need to build an electrical-circuit for the FizzBuzz game. There's a signal, called next which increment the current number by one. The rules are simple - You ... 0answers 44 views ### Trisectible Angle How do we prove that a triangle with sides (one, x, y), where x is any constructible length from one to three at the elliptic curve$$y^2 = x^3 -x^2 -x +1$$then the triangle possess at least ... 0answers 62 views ### Fermat's Last Theorem: A natural extension It is well known that there are no solutions to$$a_1^n+a_2^n=b^n$$for a_1,a_2,b\in\mathbb{Z}^+ and n>2. Is it then true that there are no solutions to$$a_1^n+a_2^n+\cdots+a_m^n=b^n$$... 0answers 71 views ### Can anyone improve on this work and find a closed form of \zeta(3)? This was something I and another user came across independently, although he decided to post it on reddit. So while its already online, let me reproduce it here with the hope that someone will be able ... 1answer 27 views ### Was this arithmetic Möbius/Mangoldt function ever used for something? Let n=\prod_k p_k^{c_k}, with p_k \in \mathbb P and$$ A(n)=\sum_{d|n} \mu(d)\Lambda(d), $$with the \mu Möbius function, which has values in {−1, 0, 1} depending on the factorization of n ... 5answers 140 views ### What is \limsup_{n\to\infty} \frac{p_{n+1}}{p_n}? Let (p_n)_{n\in\mathbb N} be the strictly increasing sequence of all primes. I'm wondering what$$S:=\limsup_{n\to\infty} \frac{p_{n+1}}{p_n}$$is. Is the result already known? By Bertrand's ... 2answers 27 views ### Proof - Uniqueness part of unique factorization theorem The uniqueness part of the unique factorization theorem for integers says that given any integer n, if n=p_1p_2 \ldots p_r=q_1q_2 \ldots q_s for some positive integers r and s and prime ... 0answers 27 views ### Limit of an euler product Before I can ask my question, I have to state a couple of definitions. Let f be a multiplicative function and let$$ D_f(s) = \sum_1^{\infty} \frac{f(n)}{n^s}, $$and define \Lambda_f(n) as ... 0answers 36 views ### Does this set contain these numbers? How would I go about proving whether or not every number n=k^8 is included in the set of all numbers m=k^4 (n and k are integers in both cases)? 2answers 33 views ### Lucas's proof of a special case of Beal's conjecture While studying the properties of a certain elliptic curve, I came across the equation x^4+y^4=z^3. There is no solution of this equation in relatively prime integers, and this is a special case of ... 4answers 124 views ### How to show that 2\times 10^{18}<20!<3 \times 10^{18} without calculator? [on hold] I want to find the first digit of 20! By calculator 20! = 2.43290200817664 \times 10^{18}. So I want to show that 2\times 10^{18}<20!<3 \times 10^{18} Thank you. 2answers 82 views ### Consecutive squarefree numbers of 5 prime factors each, mostly small The sequence of numbers 49297533, 49297534, and 49297535 is notable, because the factorizations of these numbers are each of the form a^1 \cdot b^1 \cdot c^1 \cdot d^1 \cdot e^1, where \{a\ldots ... 1answer 57 views ### How to prove\displaystyle \sum_{i=0}^{k}(-1)^i\binom{n}{k-i}\binom{n+i-1}{i}=0 I saw a combinatorial identity when i study linear-algebra, But the author didn't explain how to get it. \displaystyle \sum_{i=0}^{k}(-1)^i\binom{n}{k-i}\binom{n+i-1}{i}=0 I tried n=10 or ... 1answer 37 views ### Likelihood at least 2 out of n numbers are visible to each other in \mathbb{Z}^n Two points in  \mathbb{Z}^n  are said to be visible to each other, if they can be connected by a straight line, which doesn't intersect any points of  \mathbb{Z}^n  In Apostol's book "An ... 0answers 24 views ### On some factorial inequalities Denote P_n to be product of primes at most n. What is the minimum value of m such that P_m\geq P_n^2? What is the minimum value of m such that m!\geq n!^2? What is the minimum value of ... 5answers 69 views ### there does not exist a perfect square of the form 7\ell+3 I have been trying to prove that there does not exist a perfect square of the form 7\ell+3. I've tried using n as even or odd, and I'm getting stuck. Can someone put me on the path? Is this an ... 0answers 15 views ### Why does the uniqueness theorem for Dirichlet series hold for the infinite sums, while obviously not for partial sums? I asked in a previous question whether a function, a_n, is unique to F(s) for any Dirichlet function defined by the following$$F(s)=\sum_{n=1}^\infty{\frac{a_n}{n^s}}.$$Its uniqueness property ... 0answers 17 views ### What are the asymptotic considerations in the following? The following is from this paper that discusses polynomials and classic number theory functions. The proof of theorem 1.3 has a final statement saying that R must be null because we arrive at ... 2answers 39 views ### Number of divisors d of n^2 so that d\nmid n and d>n I just wanted to share this nutshell with you guys, it is a little harder in this particular case of the problem: Find the number of divisors d of a^2=(2^{31}3^{17})^2 so that d does not ... 2answers 45 views ### Existence of integer solution of a^2 -17b^2 =  any constant When checking whether if 9-\sqrt{17} in the ring \{a+b\sqrt17: a,b \in \mathbb{Z}\} is a prime. Suppose$$\alpha\cdot \beta = 9-\sqrt{17},$$using norm argument$$N(\alpha)N(\beta) = ...
If $a,b\in\Bbb{N}$, then what is the smallest non-trivial solution to this equation? $$\frac{\lfloor100{\sqrt[3]{a}}\rfloor}{100}+\frac{1}{100}=b$$ So I want the answer like this: ...