Articles of elementary number theory

How to prove that any natural number $n \geq 34$ can be written as the sum of distinct triangular numbers?

Sloane’s A053614 implies that $2, 5, 8, 12, 23$, and $33$ are the only natural numbers $n \geq 1$ which cannot be written as the sum of distinct triangular numbers (i.e., numbers of the form $\binom{k}{2}$, beginning $1,3,6,10,15,\ldots$). Question: How to prove that any natural number $n \geq 34$ can be written as the sum […]

System of equations modulo primes

Is it generally possible to solve a set of linear equations modulo some prime numbers $\{p,q,r\}$. For example if I have the following congruences: $$ xa_p + yb_p \equiv d \pmod {p}\\ xa_q + yb_q \equiv d \pmod {q}\\ xa_r + yb_r \equiv d \pmod {r}\\ $$ for some known $a_i$ and $b_i$ with $i […]

Is there a solution to $a^4+(a+d)^4+(a+2d)^4+(a+3d)^4+\dots = z^4$?

Is there a, $$a^4+(a+d)^4+(a+2d)^4+(a+3d)^4+\dots = z^4\tag1$$ in non-zero integers? One can be familiar with, $$31^3+33^3+35^3+37^3+39^3+41^3 = 66^3\tag2$$ I found that, $$29^4+31^4+33^4+35^4+\dots+155^4 = 96104^2\tag3$$ which has $m=64$ addends. The equation, $$a^4+(a+b)^4+(a+2b)^4+\dots+(a+63b)^4 = y^n\tag4$$ or, $$64 a^4 + 8064 a^3 b + 512064 a^2b^2 + 16257024 a b^3 + 206447136 b^4 = y^n\tag5$$ for $n=2$ can be […]

Zeros of the decimal representation of $k!$

I’d like a hint for the question: For how many positive integers $k$ does the ordinary decimal representation of the integer $k!$ end in exactly $99$ zeros? Thanks.

Regarding identities with sums of consecutive squares

This comic http://abstrusegoose.com/63 points out an interesting identity with sums of consecutive squares. Let us take positive integers $k, p, q$, with $p < q$, and ask if $k^2 + \ldots + (k + p)^2 = (k + p + 1)^2 + \ldots + (k + q)^2$. One may see this is equivalent to $(p+1)(6k^2 […]

Every two positive integers are related by a composition of these two functions?

How would one prove/disprove this? … Conjecture: Suppose $p$, $q$ are distinct primes, and define $\ f(n) = n p, \ g(n) = \left \lfloor \frac{n}{q} \right \rfloor$ for all $n \in \mathbb{N_+}$; then for all $x,y \in \mathbb{N_+}$, there exists a composition $F = g \circ g \circ \cdots \circ g \circ f \circ […]

a theorem of Fermat

While I was surfing the web, searching things about math, I read something about a particular theorem of Fermat. It said: let $a$ and $b$ be rational. Then $a^4-b^4$ cannot equal the square of a rational number, so $a^4-b^4\neq c^2$ with $c$ rational. My question is: did I understand the theorem? if not, what is […]

$n = a^2 + b^2 = c^2 + d^2$. What are the properties of a, b, c and d?

If n is a positive integer that can be represented as the sum of two odd squares in two different ways: $$ n = a^2 + b^2 = c^2 + d^2 $$ where $a$, $b$, $c$ and $d$ are discrete odd positive integers, what properties can be deduced about $a$, $b$, $c$ and $d$? There’s […]

There is no Pythagorean triple in which the hypotenuse and one leg are the legs of another Pythagorean triple.

According to Wikipedia, There are no Pythagorean triples in which the hypotenuse and one leg are the legs of another Pythagorean triple. I cannot find the proof in the citation provided. I am really struggling to proving this, can somebody please direct me to a proof (or give some advice on how to prove it, […]

Proving for $n \ge 25$, $p_n > 3.75n$ where $p_n$ is the $n$th prime.

The elements of the reduced residue system modulo $30$ are $\{1, 7, 11, 13, 17, 19, 23, 29\}$ If we order them as $e_1, e_2, e_3, \dots$ so that $e_1 = 1, e_2 = 7, \dots$, it follows that $3.75(i-1) < e_i < 3.75i$. We can generalize this. If $\gcd(x,30)=1,$ then $x = 30a + […]