# 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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### Prove that there exist $i \neq j$ with $ia_i \equiv ja_j$ mod $p$.

Let $p$ be an odd prime, and assume that $a_i \equiv i \pmod p$ for $1 \leq i \leq p-1$. Prove that there exist $i \neq j$ with $ia_i \equiv ja_j \pmod p$. This is one of my textbook problem but I ...
1answer
393 views

### In an examination the maximum marks for each of the three papers are 50 each. Maximum marks for the fourth paper are 100. …

Problem : In an examination the maximum marks for each of the three papers are 50 each. Maximum marks for the fourth paper are 100. Find the number of ways in which the candidate can score 60 % marks ...
1answer
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### Confusion on Inert Primes in Ireland and Rosen

In Ireland and Rosen, the following law for inert rational primes in a quadratic field is stated as: if $p\nmid \delta_K$, where $\delta_K$ is the discriminant of the quadratic field, and $d$ is a ...
0answers
22 views

### Need help applying euclidean algorithm to polynomials

I don't understand how the euclidean algorithm can be used to find the gcd of two polynomials in the following example: $x^3-27=(x-3)(x^2+3x+9)$ $2x^3-11x^2+16x-3=(x-3)(2x^2-5x+1)$ so we expect the ...
0answers
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### Who first proved Fermat's Last Theorem for polynomials and when?

Who first proved Fermat's Last Theorem for polynomials and when? I have a proof using the Mason-Stothers Theroem, but the result is much older. Does anyone know the original proof or prover? Or at ...
2answers
60 views

### Can two pythagoras triplet have a common number

If I have a pythagoras triplet $(a,b,c)$ such that $$a^2+b^2=c^2$$ then is there another triplet $(a,d,e)$ possible such that $$a^2+d^2=e^2, \; b\neq d$$
1answer
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### The exponential extension of $\mathbb{Q}$ is a proper subset of $\mathbb{C}$?

This question come from a recent post Exponential extension of $\mathbb{Q}$. An exponential field is a field $\mathbb{K}$ where it's well defined a function $E:\mathbb{K} \rightarrow \mathbb{K}$ ...
1answer
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### At which p-adic fields does the equation have no solution?

I have to check if the equation $3x^2+5y^2-7z^2=0$ has a non-trivial solution in $\mathbb{Q}$. If it has, I have to find at least one. If it doesn't have, I have to find at which p-adic fields it has ...
0answers
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### What is the relationship between GRH and Goldbach Conjecture?

We know that we can prove weak Goldbach Conjecture (three prime theorem) if we assume GRH (Hardy-Littlewood had proved this). Can we also prove strong Goldbach Conjecture if we assume GRH ? Also, ...
1answer
25 views

### Points where this three varible function takes the value $1$

I need to find minima of this function. $f(a,b,c)=2^a-5^b\cdot7^c$ where $a,b,c$ are positive integer I need to prove that for any value of a,b,c the value of function can never be 1. Tried ...
1answer
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### If r,s,t are prime numbers and p,q are the positive integers such that LCM of p,q is $r^2s^4t^2$ then find the …

Problem : If r,s,t are prime numbers and p,q are the positive integers such that LCM of p,q is $r^2s^4t^2$ then find the number of ordered pairs (p,q)? Can we use this : let $r^2s^4t^2$ = $2^23^45^2$...
1answer
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### Available in 6-packs, 9-packs, 20-packs

Started with an algorithms problem which says item is sold in 3 different sizes of boxes. These 3 boxes have 6, 9, 20 items each. Input is n, figure out if you can ...
0answers
40 views

### Is this “by symmetry” statement valid?

Problem: Let $p,q,r$ be integers such that $\gcd(p,q,r)=1$. Prove that there exists an integer $A$ such that $\gcd(p,q+Ar)=1$. A start: Assume for the sake of contradiction that $\gcd(p,q+Ar)>1$ ...
1answer
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### Diophantine Equation 1

I want to solve for positive integral values of $x$ and $y$: $$1216562x=87654321y+a$$ Here $a$ is a positive integer. For example if $a=40642509$ then one solution is : $x=37716$ and $y=523$ How do I ...
1answer
484 views

### Kummer-Dedekind's factorisation theorem

For a number field extension $K$ of $\mathbb{Q}$ one can factorise almost all prime ideals $(p)$ in the extension $K$, except finitely many, easily by factorising minimal polynomials in finite fields....
3answers
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### divisibility of $n^{15} - n^3$ by $32760$

I have a question & I have no idea where to begin. I hope someone here can help me. Been stuck for a while. Prove or disprove: $n^{15} - n^3$ is divisible by $32760$ for all $n \ge 0$.
1answer
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### Extended Euclid Algorithm [duplicate]

A Linear Diophantine Equation is of the following form: $Ax+By+C=0$, where $x_1\le x\le x_2$ and $y_1\le y\le y_2$. If the value of $A$, $B$, $C$, $x_1$, $x_2$, $y_1$, $y_2$ are given and $x_1\le x_2$...
0answers
24 views

### Problem on solving congruence equation 2

I want to find smallest solution $x$ for the following equation for known positive integer values $a$, $b$ and $m$. $$a-bx+2^{6x+1}\equiv0\ \pmod{m}$$ Any help will be appreciated.
2answers
159 views

### Primes for which a polynomial splits completely

Suppose that $f(x) \in \mathbb{Z}[x]$ is an irreducible polynomial over $\mathbb{Q}$. Nevertheless, it may be the case that $f(x)$ is reducible modulo $p$ for some prime $p$. What is the density of ...
2answers
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### Show that $\dfrac{2^p}{p}$ has remainder of $2$ for any prime $p \geq 3$

A bonus question on my last math exam I haven't been able to solve. Thanks for the help.
1answer
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### $\pi$ normal to the base $10$ [closed]

If $\pi$ is normal to base $10$, why would we expect to find a string of ten $0$'s in its decimal expansion?
1answer
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### Is there an Asymptotic Formula for the Largest Prime Factor of a Number?

It seems the asymptotic formula, $\sum_{n=2}^{x} P(n)$ ~ $\frac{\pi^2}{12}\frac{x^{2}}{log(x)}$ as $x \rightarrow \infty$ where $P(n)$ is the largest prime factor of the positive integer $n$, cannot ...
2answers
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### Are square numbers also known as rectangular numbers, too?

I've learned that a square number can be multiplied by itself twice and be used as a base raised to the exponent of $2$, such as $3^2, 9^2, 0^2,$ etc. Square numbers can also fit into square shapes. ...
1answer
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### How is it possible that two and three are the only consecutive prime numbers?

I've learned that two and three are not only two consecutive numbers, but are also two consecutive prime numbers. How is this possible? I think I'm on the right track in the following text: The ...
1answer
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1answer
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