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Comparing two numbers written in index form like 2^{11} and 3^7 is not difficult, as any calculator would confirm that 2

A program written in the programming language Fractran consists of a list of fractions.

Consider quadratic Diophantine equations of the form: For example, when D=13, the minimal solution in x is 649^2 - 13 ti

A positive integer, n, is a near power sum if there exists a positive integer, k, such that the sum of the kth powers of

Alice and Bob have enjoyed playing Nim every day.

The number sequence game starts with a sequence S of N numbers written on a line.

Let d(p, n, 0) be the multiplicative inverse of n modulo prime p, defined as n times d(p, n, 0) = 1 bmod p.

A line segment of length 2n-3 is randomly split into n segments of integer length (n ge 3).

In the following equation x, y, and n are positive integers.

Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for examp

Turan has the electrical water heating system outside his house in a shed.

Consider the triangle with sides sqrt 5, sqrt {65} and sqrt {68}.

Consider the divisors of 30: 1,2,3,5,6,10,15,30.

Alice and Bob are taking turns playing a game consisting of c different coins on a chessboard of size n by n.

Let S(n,m) = sumphi(n times i) for 1 leq i leq m.

Let p(t) denote the (t+1)th prime number.

For two integers n,e gt 1, we define an (n,e)-MPS (Mirror Power Sequence) to be an infinite sequence of integers (ai){ig

Given is the arithmetic-geometric sequence u(k) = (900-3k)r^{k - 1}.

Let f be a function from a finite set S to itself.

A positive integer N is stealthy, if there exist positive integers a, b, c, d such that ab = cd = N and a+b = c+d+1.

Consider the Gaussian integer i-1.

The cube, 41063625 (345^3), can be permuted to produce two other cubes: 56623104 (384^3) and 66430125 (405^3).

Consider the following set of dice with nonstandard pips: Die A: 1 4 4 4 4 4 Die B: 2 2 2 5 5 5 Die C: 3 3 3 3 3 6 A gam

For any prime p the number N(p, q) is defined by N(p, q) = sum{n = 0}^q Tn cdot p^n with Tn generated by the following r

Let Sm = (x1, x2, dots , xm) be the m-tuple of positive real numbers with x1 + x2 + cdots + xm = m for which Pm = x1 cdo

Let f(n) be the number of couples (x, y) with x and y positive integers, x le y and the least common multiple of x and y

Let y0, y1, y2, dots be a sequence of random unsigned 32-bit integers (i.e.

Let d(k) be the sum of all divisors of k.

The (a,b,m)-sequence, where 0 leq a,b lt m, is defined as $begin{align} g(0)&=a g(1)&=b g(n)&= big(g(n-1) + g(n-2)big) b

A triplet of positive integers (a, b, c) is called a Cardano Triplet if it satisfies the condition: For example, (2,1,5)

You probably know the game Fifteen Puzzle.

A positive integer n is considered cube-full, if for every prime p that divides n, so does p^3.

Consider the following "magic" 3-gon ring, filled with the numbers 1 to 6, and each line adding to nine.

Working from left-to-right if no digit is exceeded by the digit to its left it is called an increasing number; for examp

A Hamming number is a positive number which has no prime factor larger than 5.

Consider the problem of building a wall out of 2 times 1 and 3 times 1 bricks (text{horizontal} times text{vertical} dim

Circles A and B are tangent to each other and to line L at three distinct points.

Bob is very familiar with the famous mathematical puzzle/game, "Tower of Hanoi," which consists of three upright rods an

Given a set, L, of unique lines, let M(L) be the number of lines in the set and let S(L) be the sum over every line of t

Let S(A) represent the sum of elements in set A of size n.

We want to tile a board of length n and height 1 completely, with either 1 times 2 blocks or 1 times 1 blocks with a sin

The primes 3, 7, 109, and 673, are quite remarkable.

Tom has built a random generator that is connected to a row of n light bulbs.

A game is played with two piles of stones and two players.

Let S(n, k, b) represent the number of valid solutions to x1 + x2 + cdots + xk le n, where 0 le xm le b^m for all 1 le m

It can be shown that the polynomial n^4 + 4n^3 + 2n^2 + 5n is a multiple of 6 for every integer n.

By starting at the top of the triangle below and moving to adjacent numbers on the row below, the maximum total from top

Let a1, a2, dots, an be an integer sequence of length n such that: - a1 = 6 - for all 1 le i lt n: phi(ai) lt phi(a{i +

ABC is an integral sided triangle with sides a le b le c.

Let g(m) be the integer defined by the following double sum of products of binomial coefficients: You are given that g(1

Given is the function f(a,n)=lfloor (lceil sqrt a rceil + sqrt a)^n rfloor.

There are 16 positive integers that do not have a zero in their digits and that have a digital sum equal to 5, namely: 5

In the following equation x, y, and n are positive integers.

A positive integer with k (decimal) digits is called balanced if its first lceil k/2 rceil digits sum to the same value

An integer s is called a superinteger of another integer n if the digits of n form a subsequenceA subsequence is a seque

Let P{a,b} denote a path in a atimes b lattice grid with following properties: - The path begins at (0,0) and ends at (a

We define a pseudo-geometric sequence to be a finite sequence a0, a1, dotsc, an of positive integers, satisfying the fol

For a positive integer, n, define g(n) to be the maximum perfect square that divides n.

Given an irrational number alpha, let Salpha(n) be the sequence Salpha(n)=lfloor {alpha cdot n} rfloor - lfloor {alpha c

We define an S-number to be a natural number, n, that is a perfect square and its square root can be obtained by splitti

For a set of positive integers a, a+1, a+2, dots , b, let C(a,b) be the number of non-empty subsets in which the product

A composite number can be factored many different ways.

For a non-negative integer k, the triple (p,q,r) of positive integers is called a k-shifted Pythagorean triple if (p, q,

By counting carefully it can be seen that a rectangular grid measuring 3 by 2 contains eighteen rectangles: Although the

A number consisting entirely of ones is called a repunit.

It is possible to find positive integers A and B such that given any triangular number, Tn, then ATn +B is always a tria

For an integer n ge 4, we define the lower prime square root of n, denoted by operatorname{lps}(n), as the largest prime

The points P(x1, y1) and Q(x2, y2) are plotted at integer co-ordinates and are joined to the origin, O(0,0), to form tri

Each one of the 25 sheep in a flock must be tested for a rare virus, known to affect 2 of the sheep population.

For a positive integer n, the function g(n) is defined as For example, g(4) = -gcd left(4,1^2right) + gcd left(4,2^2righ

We are trying to find a hidden number selected from the set of integers 1, 2, dots, n by asking questions.

Let C(n) be the number of squarefree integers of the form x^2 + 1 such that 1 le x le n.

We use xoplus y for the bitwise XOR of x and y.

The harmonic series 1 + frac 1 2 + frac 1 3 + frac 1 4 + cdots is well known to be divergent.

Consider the number 48.

We define the Matrix Sum of a matrix as the maximum possible sum of matrix elements such that none of the selected eleme

In laser physics, a "white cell" is a mirror system that acts as a delay line for the laser beam.

Let T(n) be the number of tours over a 4 times n playing board such that: - The tour starts in the top left corner.

An electric circuit uses exclusively identical capacitors of the same value C.

It can be seen that the number, 125874, and its double, 251748, contain exactly the same digits, but in a different orde

A list of size n is a sequence of n natural numbers.

A touch-screen device can be unlocked with a "password" consisting of a sequence of two or more distinct spots that the

A positive number is pandigital in base b if it contains all digits from 0 to b - 1 at least once when written in base b

The lambda-calculus is a universal model of computation at the core of functional programming languages.

Let T(n, m) be the number of m-tuples of positive integers such that the sum of any two neighbouring elements of the tup

Every triangle has a circumscribed circle that goes through the three vertices.

N disks are placed in a row, indexed 1 to N from left to right.

We call the convex area enclosed by two circles a lenticular hole if: - The centres of both circles are on lattice point

Given a character string s, we define L(k,s) to be the length of the longest substring of s which appears at least k tim

The well-known Rubik's Cube puzzle has many fascinating mathematical properties.

Find the number of integers 1 lt n lt 10^7, for which n and n + 1 have the same number of positive divisors.

The number 7 is special, because 7 is 111 written in base 2, and 11 written in base 6 (i.e.

Consider the following binary quadratic form: A positive integer q has a primitive representation if there exist positiv

70 coloured balls are placed in an urn, 10 for each of the seven rainbow colours.

Define f(0)=1 and f(n) to be the number of ways to write n as a sum of powers of 2 where no power occurs more than twice

A permutation of 2,3,ldots,n is a rearrangement of these numbers.

Let f(n) be the largest prime factor of n and displaystyle F(n) = sum{i=2}^n f(i).

The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways: (i) eac

Using only a six-sided fair dice and a five-sided fair dice, we would like to emulate an n-sided fair dice.

A laborious ant walks randomly on a 5 times 5 grid.