Articles of elementary number theory

If $\sqrt{a} + \sqrt{b}$ is rational then prove $\sqrt{a}$ and $\sqrt{b}$ are rational

Assume there exist some rationals $a, b$ such that $\sqrt[3]{a}, \sqrt[3]{b}$ are irrationals, but: $$\sqrt[3]{a} + \sqrt[3]{b} = \frac{m}{n}$$ for some integers $m, n$ $$\implies \left(\sqrt[3]{a} + \sqrt[3]{b}\right)^3 = \frac{m^3}{n^3}$$ $$\implies a + b + 3 \cdot \sqrt[3]{ab}\left(\sqrt[3]{a} + \sqrt[3]{b}\right) = \frac{m^3}{n^3}$$ Since $a +b$, $\sqrt[3]{a} + \sqrt[3]{b}$ are rational, $\sqrt[3]{ab}$ must be rational as […]

Prove that for no n>1 is the sum $(1!)^2+\cdots+(n!)^2$ a perfect square.

Prove that for no $n>1$ is the sum $(1!)^2+\cdots+(n!)^2$ a perfect square. I know that $(1!)^2+\cdots+(n!)^2=\sum_{j=1}^{n} (j!)^2$. I also know that the all squares are congruent to 0 or 1 mod 4. But I was experimenting with a couple n’s and found that they are equal to 1 mod 4. Is the statement above true?

$p$ prime, $1 \le k \le p-2$ there exists $x \in \mathbb{Z} \ : \ x^k \neq 0,1 $ (mod p)

I found this problem in my algebra book, but unfortunately, there is no solution included. Here it is: Let $p$ be a prime, $1 \le k \le p-2$. Show that there exists $x \in \mathbb{Z} \ $ such that $\ x^k \neq 0,1$ (mod $p$). It looks like a very nice problem, but I have […]

Eulers totient function divided by $n$, counting numbers in the set that are coprime to n

If we divide Euler’s totient function $\omega(n)$ by $n$, we obtain a fraction. If we multiply this fraction by any natural number $m$ which gives us another natural number $p$, is it true that $p$ is the number of natural numbers in $[1,m]$ that are coprime to $n$?

Roman Numbers – Conversion to decimal number

I have read that if a smaller number is to the left of a larger number means that the smaller number has to be subtracted from the larger number. Ok I can understand quickly for below Roman Numbers : IX = X – I = 10 – 1 = 9 But I have difficulty in […]

Prove that $(ab,cd)=(a,c)(b,d)\left(\frac{a}{(a,c)},\frac{d}{(b,d)}\right)\left(\frac{c}{(a,c)},\frac{b}{(b,d)}\right)$

I’m working through Oystein Ore’s Number Theory and its History. On p. 109, I’m stuck on #2. The question asks the reader to verify the following identity [Note: $(x,y)=\gcd(x,y)$]: $$(ab,cd)=(a,c)(b,d)\left(\frac{a}{(a,c)},\frac{d}{(b,d)}\right)\left(\frac{c}{(a,c)},\frac{b}{(b,d)}\right)$$ I’ve tried numerous numeric examples and not found an exception. I’ve tried a messy proof, substituting sample factors and exponents, but it’s not very cohesive, […]

Using Fermat's Little Theorem, find the least positive residue of $3^{999999999}\mod 7$

I’m going through the problems in Rosen’s Elementary Number Theory and am having some trouble with the this problem, Find the least positive residue of $3^{999999999}\mod 7$.

Showing there exists infinite $n$ such that $n! + 1$ is divisible by atleast two distinct primes

This is a homework question. I know there exists infinitely many primes. Let $n = p-1$ and so by Wilson’s theorem we know there exists atleast one prime $p$ that divides $n! + 1$. I used wolframalpha and checked for a couple of $n = p-1$ values and all show me that there are in […]

Is it impossible to recover multiplication from the division lattice categorically?

In this question it was asked if the division lattice (i.e., the preorder category $(\Bbb Z_{>0}, \mid)$) contains enough information categorically to recover the relation $ab=\gcd(a,b)\operatorname{lcm}(a,b)$. I answered that this is impossible because the division lattice does not know what multiplication is; we first have to introduce it as a tensor product. However, while it […]

Definition of gcd in Euclidean domain dependent on valuation?

Let $D$ be Euclidean. In $\mathbb Z[i]$ our lecturer defined $\gcd (a,b)$ as the common divisor with greatest norm. I have a slight problem with this, since we actually fix the norm $N(a+bi)=a^2+b^2$. Does this definition indeed depend on the norm or is it actually equivalent to the general definition via the universal property $$c\mid […]