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Browse High School Number Theory
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Diophantine equations.
    Infinite number of primes?
    Testing for primality.
    What is 'mod'?

Running Time of an Insertion Sort [08/06/1999]
How can I write and solve a recurrence formula for the running time of an insertion sort? Which is better, an insertion sort or a merge-sort?

Search for the Largest Prime [08/01/2000]
What is the largest finite number that has a practical use in some branch of mathematics or science? What is the largest prime number known?

Second-Degree Two-Variable Diophantine Equation [04/12/2001]
Solve Ax^2+Bxy+Cy^2+Dx+Ey+F = 0 where B^2-4AC=k^2 for some integer k.

Second-Order Linear Recurrences [06/08/2001]
Three problems involving recurrence equations.

Second Order Recurrence with Non-Constant Coefficients [05/27/2005]
I'm trying to find a closed form solution of a second order recurrence relation with no constant coefficients, specifically: u(n+2) = 2*(2*n+3)^2 * u(n+1) - 4*(n+1)^2*(2*n+1)*(2*n+3)*u(n). Can you help?

Semitonal Half-Stepping, Ever Sharper [05/05/2012]
An octogenarian wonders why modulating through the circle of fifths results in adjacent scales that become successively sharper. Doctor George applies some modular arithmetic to peel away at the chromatic scale — then spontaneously augments his original response to offer clearer insight.

Sequence of Integers [08/12/2008]
Find all functions f such that for each n in Z+ we have f(n) > 1 and f(n + 3)f(n + 2) = f(n + 1) + f(n) + 18.

Set Theory and GCD and Divisibility [01/27/2003]
If 1 <= a <= n and 1 <= b <= n and ab <= n, and if a divides n and b divides n, does that mean that ab divides n given that GCD(a,b) = 1 ?

Show 2^(N-1) Congruent to 1(mod N) [02/25/2003]
I need to show that if N = 2^p - 1, p prime, then 2^(N-1) is congruent to 1(mod N).

Showing a Diophantine Equation Has No Solutions [07/30/2008]
Do there exist positive integers m and n such that m^3 = 3n^2 + 3n + 7?

Showing Divisibility [07/12/1998]
How do you show that 5^(2n) + 3(2^(2n+1)) is divisible by 7?

Showing Two Numbers Are Relatively Prime [08/01/2008]
Show that for every natural number n, 21n + 4 and 14n + 3 are relatively prime.

Show n^3 + 11n Divisible by 6 [12/11/2002]
If n is a natural number, show that for all values of n, (n^3+11n) is divisible by 6.

Sigma Notation [12/17/1998]
Some summation formulas; finding Sum((n+1)^2).

Significance of Irrational Numbers [08/23/1999]
What exactly is the meaning of .333... or pi? What's the difference between point three repeating and point three to the 105th decimal place?

Simple Example of Ramanujan's Work [03/28/1999]
Ramanujan's contributions to the divisibility properties of partitions of whole numbers.

Simultaneous Modulus Congruencies [04/18/2001]
How can I find x if x = 3 (mod 8), x = 11 (mod 20) and x = 1 (mod 15)?

Sizes of Infinities [01/31/1997]
How can you prove that one infinity is larger than another?

Smallest Number Puzzle [06/30/1998]
Find the smallest number which when divided by 9,13,17, and 25 leaves remainders 1,0,2, and 3 respectively.

The Smallest Number Which When Divided Leaves Specific Remainders [02/27/2010]
Doctor Rick, an eleven year-old, and her father apply least common multiples, modular arithmetic, and the Chinese Remainder Theorem to reason their way to the smallest number which when divided by 3, 7, and 11 leaves remainders 1, 6, and 5, respectively.

A Solution in Natural Numbers [10/30/2001]
Prove that x^2+y^2=z^n has a solution in natural numbers for all n, where n is a natural number.

Solving a Diophantine Equation [05/01/2005]
How can I find all integer solutions of an equation in the form aXY + bX + cY + d = 0? For example, 5XY + 3X - 8Y - 8 = 0.

Solving a Diophantine Equation [01/28/2005]
Solve the following Diophantine equation: 5x + 3kx = 8k^2 - 25

Solving Cubic and Quartic Polynomials [04/30/1998]
Could you describe the algorithms used to solve cubic and quartic polynomials (Tartaglia's Solution)?

Solving Diophantine Equations By Organized Thinking [11/25/2003]
Laura is in charge of lighting. Each light fixture supplies exactly 1,000 watts of power to light the bulbs in the fixture. Laura can use any combination of 150-watt, 100-watt, 75-watt, or 60-watt bulbs, but the total number of watts must be 1,000. How many different combinations of bulbs could Laura use in a light fixture?

Solving Modular Formula [11/04/2008]
If a = b^e (mod c), how do I solve for b if I know a and c?

Solving Multivariable Diophantine Equations [05/03/1998]
Finding general solutions to two diophantine equations.

Solving the Diophantine Equation x^y - y^x = x + y [04/30/2005]
Find all integer solutions of x^y - y^x = x + y.

Solving x^y = y^(x - y) for All Natural x, y [11/18/2010]
A student seeks all natural numbers x, y such that x^y = y^(x - y). With chains of reasoning about integer divisibility and exponentiation, Doctor Vogler deduces all three solutions.

Spacing between Prime Numbers [11/08/2005]
Where is the first place that the difference between two consecutive prime numbers exceeds 2000? Is there a formula or general approach to finding such differences without having to just read through lists of known primes?

Splitting a Sum of Integers into Two Equal Sums [09/07/2004]
Given integers {1,2,3,4,...n}, prove that if their sum is even they can be split into two equivalent sums, each equal to half of the original sum, and using each integer once.

The Square and Multiply Method [08/31/1998]
Solve an encryption problem by solving the math function 33815^(81599) (mod 154381).

Square Numbers, All Digits the Same [06/17/2003]
Is there any square number with all the same digits?

Square of an Odd Number [11/12/2002]
True or false: the square of any odd number can be represented in the form 8n+1, where n is a whole number.

Square Root and Sum of Digits [07/25/2008]
I noticed that 81 has the same square root as the sum of its digits since 8 + 1 = 9 and the square root of 81 is also 9. Are there other numbers that have the same property?

Square Root of 2 as a 'Vulgar Fraction' [05/04/2001]
Can the square root of 2 be expressed as a fraction?

Square Root of a Prime [07/14/1999]
Suppose p is a prime number. Show that sqrt(p) is irrational.

The Square Root of i [05/25/1997]
What is the square root of i?

The Square Root of n! [10/14/1998]
For what natural numbers n is the square root of n! an integer?

Square Roots in Binary [10/03/2000]
Can you show an example of taking the square root of a binary number?

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