Number Theory

by Fasih, Apr 18, 2025, 6:44 PM

Find all integer solutions of the equation $x^{3} + 2 ^{\text{y}} = p^{2}$ for all x, y $\ge$ 0, where $p$ is the prime number.

author @Fasih

This post has been edited 1 time. Last edited by Fasih, 2 hours ago
Reason: typo

number theory

Integers

prime numbers

Polynomial functional equation

by Fishheadtailbody, Apr 18, 2025, 6:03 PM

P(x) is a polynomial with real coefficients such that
P(x)^2 - 1 = 4 P(x^2 - 4x + 1).
Find P(x).

Click to reveal hidden text

algebra

polynomial

functional equation

Arrangement of integers in a row with gcd

by egxa, Apr 18, 2025, 5:09 PM

Let $n$ be a natural number. The numbers $1, 2, \ldots, n$ are written in a row in some order. For each pair of adjacent numbers, their greatest common divisor (GCD) is calculated and written on a sheet. What is the maximum possible number of distinct values among the $n - 1$ GCDs obtained?

This post has been edited 1 time. Last edited by egxa, 5 hours ago

GCD

number theory

greatest common divisor

powers sums and triangular numbers

by gaussious, Apr 17, 2025, 1:00 PM

prove 1^k+2^k+3^k + \cdots + n^k \text{is divisible by } \frac{n(n+1)}{2} \text{when} k \text{is odd}

number theory

Hard Polynomial

by ZeltaQN2008, Apr 16, 2025, 2:38 PM

Let ?(?) be a polynomial with integer coefficients. Suppose there exist infinitely many integer pairs (?,?) such that
?(?) + ?(?) = 0. Prove that the graph of ?(?) is symmetric about a point (i.e., it has a center of symmetry).

Grasshoppers facing in four directions

by Stuttgarden, Mar 31, 2025, 1:12 PM

Let $S$ be a finite set of cells in a square grid. On each cell of $S$ we place a grasshopper. Each grasshopper can face up, down, left or right. A grasshopper arrangement is Asturian if, when each grasshopper moves one cell forward in the direction in which it faces, each cell of $S$ still contains one grasshopper.

Prove that, for every set $S$ , the number of Asturian arrangements is a perfect square.
Compute the number of Asturian arrangements if $S$ is the following set:

Attachments:

combinatorics

Spain

f(x+y+f(y)) = f(x) + f(ay)

by the_universe6626, Feb 21, 2025, 1:16 PM

For a given integer $a$ , find all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ such that
$f(x+y+f(y))=f(x)+f(ay)$ holds for all $x,y\in\mathbb{Z}$ .

(Proposed by navi_09220114)

algebra

functional equation

complex bashing in angles??

by megahertz13, Nov 5, 2024, 2:35 AM

Let $\gamma$ and $I$ be the incircle and incenter of triangle $ABC$ . Let $D$ , $E$ , $F$ be the tangency points of $\gamma$ to $\overline{BC}$ , $\overline{CA}$ , $\overline{AB}$ and let $D'$ be the reflection of $D$ about $I$ . Assume $EF$ intersects the tangents to $\gamma$ at $D$ and $D'$ at points $P$ and $Q$ . Show that $\angle DAD' + \angle PIQ = 180^\circ$ .

geometry

complexbash

Bijection on the set of integers

by talkon, Apr 9, 2018, 3:12 PM

Determine all bijections $f:\mathbb Z\to\mathbb Z$ satisfying
$f^{f(m+n)}(mn) = f(m)f(n)$ for all integers $m,n$ .

Note: $f^0(n)=n$ , and for any positive integer $k$ , $f^k(n)$ means $f$ applied $k$ times to $n$ , and $f^{-k}(n)$ means $f^{-1}$ applied $k$ times to $n$ .

Proposed by talkon

algebra

functional equation

a, b subset

by MithsApprentice, Oct 22, 2005, 11:55 PM

Determine (with proof) whether there is a subset $X$ of the integers with the following property: for any integer $n$ there is exactly one solution of $a + 2b = n$ with $a,b \in X$ .

number base

Submit

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math_exploration

by Fasih, Apr 18, 2025, 6:44 PM

by Fishheadtailbody, Apr 18, 2025, 6:03 PM

by egxa, Apr 18, 2025, 5:09 PM

by gaussious, Apr 17, 2025, 1:00 PM

by ZeltaQN2008, Apr 16, 2025, 2:38 PM

by Stuttgarden, Mar 31, 2025, 1:12 PM

by the_universe6626, Feb 21, 2025, 1:16 PM

by megahertz13, Nov 5, 2024, 2:35 AM

by talkon, Apr 9, 2018, 3:12 PM

by MithsApprentice, Oct 22, 2005, 11:55 PM