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 […]

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, $$a^4+(a+d)^4+(a+2d)^4+(a+3d)^4+\dots = z^4\tag1$$ 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 […]

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.

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 […]

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 […]

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 […]

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 […]

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, […]

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 + […]

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