Articles of number theory

Quartic polynomial taking infinitely many square rational values?

I was wondering whether the value of $$P(x)=x^4-6x^3+9x^2-3x,$$ is a rational square for infinitely many rational values of $x$. Is there a general method to check this for a polynomial (in one variable)? If not, how would one go about figuring this out?

Largest Triangular Number less than a Given Natural Number

I want to determine the closest Triangular number a particular natural number is. For example, the first 10 triangular numbers are $1,3,6,10,15,21,28,36,45,55$ and thus, the number $57$ can be written as $$57=T_{10}+2$$ The number $54$ can be written as $$54=T_{9}+9\neq T_{10}-1$$ The second part highlights that I am looking for Triangular numbers larger than a […]

Number Theory Prime Reciprocals never an integer

I’m in number theory and I currently have these problems assigned as homework. I’ve looked through the sections containing these problems and I’ve solved/proved most of the other problems, but I can’t figure these ones out. For $n>1$, show that every prime divisor of $n!+1$ is an odd integer that is greater than $n$. Assuming […]

Given $x^3$ mod $55$, find its inverse

So i am wondering how i can figure out what the functional inverse of $x^3$ mod $55$ is. I can only assume it is $x^{1/3}$ mod $55$ but i am not sure if that is the form i should keep it in

Proof of $ \phi(n) = \sum_{n|d} \mu(d) \cdot\frac nd $

I’d like to prove $\phi(m)=\sum_{m|d}\mu(d)\cdot\frac md$. If I’m right then we have for euler-phi $\phi(n) = \sum_{m \leq n,\gcd(m,n)=1} 1$ Which means: as long as $m$ is less or equal than $n$ and their greatest common divisor is $1$, $\phi$ does count $1$ and sums up all “$1$”. Should be right so far. Maybe we […]

natural solutions for $9m+9n=mn$ and $9m+9n=2m^2n^2$

Please help me find the natural solutions for $9m+9n=mn$ and $9m+9n=2m^2n^2$ where m and n are relatively prime. I tried solving the first equation in the following way: $9m+9n=mn \rightarrow (9-n)m+9n=0 $ $\rightarrow m=-\frac{9n}{9-n}$ Thanks in advance.

Rational Point in circle

How many rational point(s) (a point (a, b) is called rational, if a and b are rational numbers) can exist on the circumference of a circle having centre (pie, e)

The form $xy+5=a(x+y)$ and its solutions with $x,y$ prime

In another question I was asking if there are any different $x,y>2$ primes such that $xy+5=a(x+y)$. Where $a=2^r-1$, and $r>2$. I was thinking if it is able to find a Pell equation or a similar pattern of $xy+5=a(x+y)$ to say what are and how many integer solutions are there (in particular prime solutions). Thanks.

Archimedean places of a number field

Let $K$ be a number field with an Archimedean absolute value $|\cdot |$ and let $\bar{K}$ be the completion of $K$ wrt this valuation. Then $\bar{K}\cong \mathbb R $ or $\mathbb C$. My question is: Does this imply that the Archimedean places of $K$ correspond bijectively to the real embeddings $K\hookrightarrow \mathbb R$ and complex […]

Prove that $a^2 + 1$ cannot have prime factor of the form $4k + 3$

What I have done: (not sure if it’s right) $a^2 + 1\equiv 1\pmod 4$ or $2\pmod 4$ But if it has two prime factors in the form $4k + 3$, it will be $1\pmod4$, and I don’t know where to go from here