Is the following statement true or false?Let $\mathbb{N}$ be the set of positive integers. For any $z > 2$, there always exist $x, y < z$ such that:$$f(x) + f(y) = f(z)$$Where the arithmetic function $f(n)$ is defined as:$$f(n) = \prod_{p^k \parallel n} \left( \frac{p^{k+1}-1}{p-1} \right) =...

The discovery by Andrew Booker of an integer solution to, $$N=x^3 + y^3 +z^3=33$$ $$8866128975287528^3 - 8778405442862239^3 -2736111468807040^3=33$$ got some press and Youtube mileage back in 2019. As mentioned in Booker's July 2019 article, for $0<N<1000$, there used to be $13$ unsolved $N$,...

I am looking for a reference—either a book or a specific paper—that contains the actual proof of the result that no three consecutive positive integers are perfect powers. While reading Wacław Sierpiński's 250 Problems in Elementary Number Theory, I came across a remark stating that A. Mąkowski...

I ran a simulation where for each 2-digit combination with 30 symbols (so 0 to T), it checked, from base 2 to base 10,000, in how many bases that specific symbol combination resulted in a prime number. The top 10 were 65,6B,6H,6T,6N,61,6D,67,6J, and 6P. All starting with 6. Anyone have any idea...

Let P(n) and Q(n) be two non-trivial polynomials in n with rational coefficients and z[P, Q] is the value of infinite sum of P(n)/Q(n) from n=1 to +∞ (only when it converges, in which the degree of Q should be larger than or equal to the degree of P plus 2). Claim: It is impossible for z[P,Q] to...

Background The triangular grid places integer $T(r-1)+c$ at row $r$, column $c$, where $T(n)=n(n+1)/2$. Transposing this grid, reading along SE diagonals of the triangular grid as columns, yields a new array whose column $d$ has values $$f_d(n) = T(n+d-2)+n = \frac{n^2+(2d-1)n+(d-1)(d-2)/2 + ......

About prime factorization of consecutive integers, we all can notice prime factors vary apparently without any logic. Some numbers like $82 = 2 \times 41$ have highly unequal factors (high variance among the factors), while others like $80 = 2^4 \times 5$ or $2310 = 2 \times 3 \times 5 \times 7...

Given a multiplicative function $f$ with divisor bound $|f|\le \tau_k$, where $k$ is a nonnegative real number. We consider the Dirichlet series $$ F(s)=\sum_{n=1}^\infty \dfrac{f(n)}{n^s}. $$ Since $$ \sum_{n=1}^\infty \dfrac{\tau_k(s)}{n^s}=\zeta(s)^k, $$ $F(s)$ absolutely converges on the...

Consider affine families $$ Q(u)=2^t3^{16}u+B, $$ with $t\ge 3$ and $2^t\mid 3B-1$. Set $$ C_0=\frac{3B-1}{2^t}. $$ Then $$ 3Q(u)-1 =2^t3^{17}u+(3B-1) =2^t(3^{17}u+C_0). $$ Suppose we restrict to a subfamily where, after the fixed factor $2^t$, another $2^\lambda$ divides the remaining factor:...

Numbers that are not yet mapped to are marked prime and given their own "trees", but really they are distance $\infty$ from the other primes. Traditionally, in a connected graph, BFS forms one tree, but really this is a collection of overlapping trees. What we have on the number line is a...

I apply the iterated forward difference operator to the sequence of primes; from $$ (p_n) = (2, 3, 5, 7, 11 \ldots) $$ I get $$ (d^1_n) = (1, 2, 2, 4 \ldots) $$ $$ (d^2_n) = (1, 0, 2 \ldots) $$ $$ (d^3_n) = (-1, 2 \ldots) $$ $$ \ldots $$ For each sequence $(x_n)$, one can reconstruct the...

I was solving this CSES question called Sum of Divisors and one of the solutions in the USACO Guide hints at this statement: The minimum value of $j > i$ such that $\lfloor n/j \rfloor$ < $\lfloor n/i \rfloor$ is $j = \lfloor n/q \rfloor + 1$, where $q = \lfloor n/i \rfloor$ for integers $j, i...