Articles of number theory

Can the equation $ax^2+by^2=cz^2$ be solved in integers (excluding trivial solutions)?

Suppose $a,b,c\in\mathbb{N}$ and are each squarefree. Is there a general solution for this equation? I found that for this equation to be soluble in integers there are three necessary and sufficient conditions which are $ab,bc$ and $ac$ should be quadratic residues $\bmod c,\bmod a$ and $\bmod b$ respectively. That is, the equations $$ ab\equiv\alpha^2 \hspace{-0.8em}\pmod{c} […]

special values of zeta function and L-functions

I was reading in some lectures notes about the Riemann zeta-function which takes on special values: $$\zeta(2) = \sum \frac{1}{n^2} = \frac{\pi^2}{6}$$ In fact, we can compute even values of the zeta function $\zeta(2k) \in \pi^{2k}\mathbb{Q}$. We can twist with a quadratic character and get more special values that way: $$ L(\chi_4, 2k+1) = \sum_{n […]

How generalize the alternating Möbius function?

Here is what I want to do, I have this matrix: $$\displaystyle T = \begin{bmatrix} +1&+1&+1&+1&+1&+1&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&-2&+1&+1&-2&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&+1&+1&-4&+1&+1 \\ +1&-1&-2&-1&+1&+2&+1 \\ +1&+1&+1&+1&+1&+1&-6 \end{bmatrix}$$ that I have posted many times before, and which has the definition: $$T(n,k)=a(GCD(n,k))$$ where $a$ is the Dirichlet inverse of the Euler totient function and $GCD(n,k)$ […]

How many numbers from $1$ to $99999$ have a digit-sum of $8$?

How many numbers from $1$ to $99999$ have a digit-sum of $8$? Why is the answer ${8+4\choose 4}$? Does the following method work? Answer is the number of ways to split 8 into 5 digits, i.e. number of ways to insert 4 lines among a row of 8 objects.

Bound for divisor function

I have been searching for a bound of the divisor function $d(n)$, meaning the number of divisors of n. So far I have found that it can be bounded by $$ d(n) \le e^{O(\frac{\log n}{\log \log n})}$$ Wigert has proven the constant is $\log 2$ so $$ d(n) \le e^{(\log 2+ o(1)) \frac{\log n}{\log \log […]

Summation of an array of numbers that grows quadratically in both dimensions

(Note: This question was inspired by Sum a series of series where each value increments by one, and in particular by the discussion in the comments under the answer at Lets say we have this 4×4 set of series of numbers: (1, 3, 6, 10) (2, 5, 9, 14) (4, 8, 13, 19) (7, […]

Congruence between Bernoulli numbers

I fell on the following fact as a by-product: If p is a prime number, then the sum of the Bernoulli numbers, from index 0 to p – 2, is congruent with – 1 modulo p. Do you know a simple proof ? Or a reference to the literature ? Thanks beforehand. Please, don’t latexify […]

Sequences containing infinitely many primes

What are some interesting sequences that contain infinitely many primes? If it takes form of a polynomial, Dirichlet’s theorem answer the question completely for linear polynomial. What about polynomials of degree more than 1? Is there a known polynomial of degree more than 1 that contains infinitely many primes? What about more complicated sequences like […]

Compute $v_2\left(2005^{2^{100}}-2003^{2^{100}}\right)$

Compute $v_2\left(2005^{2^{100}}-2003^{2^{100}}\right)$ where $v_2(n)$ is the largest power of $2$ dividing $n$. I think one way to solve this is to use the binomial theorem with $2005=2003+2$, but you have to know something about the largest power of two dividing binomial coefficients. I’m not sure if that behaves nicely, and its not strictly increasing with […]

Sequence generated by $2^k-1$ contains new prime factors

I was playing around with the sequence where the $k^{th}$ number is equal to $2^k-1$. It seems that all numbers except $63$ contain at least one new prime in there prime factorization. That is a prime that has not occurred in the prime factorization of any of the previous numbers in the sequence. Is there […]