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High School Archive

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'?

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Pythagorean Triple with 71 [12/07/1997]

Is there a Pythagorean triple that contains the number 71?

Quadratic Residues [06/30/1998]

I need a fundamental explanation of the concept of quadratic residues.

Quadratic Residues and Sums of Squares [10/28/1998]

In one of the lemmas in number theory, if p is an odd prime number, then there exist x, y such that x^2+y^2+1=kp...

A Quartic Diophantine Equation: 10657 + 11579x^2 + x^4 = y^2 [12/29/2008]

Doctor Vogler helps a student look for integer solutions to a quartic polynomial by noticing a difference of squares in its coefficients and factoring its constant term.

Ramsey's Theorem and Infinite Sequence [06/01/1999]

Ramsey's Theorem applied to divisibility in infinite sequences.

Rational and Irrational Numbers: Multiplication, Division [10/15/2001]

I would like the rules explained for: irrational * irrational; rational * rational; irrational/rational.

Real and Rational Numbers [02/27/2001]

How can I show that the number of rational numbers between 0 and 1 is the same as the number of natural numbers (considering the ordering of fractions: 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5...)?

Real Numbers [08/08/1997]

What exactly is a real number?

Reasoning about Integers [10/06/2004]

When positive integers p and q are divided by an even positive integer t, they have remainders 2 and t/2, respectively. What is the remainder when the product pq is divided by t?

Reciprocals of Integers Greater Than 1 as Sum of a Series [07/01/2004]

Show that the reciprocal of every integer greater than 1 is the sum of a finite number of consecutive terms of the series 1/[j(j + 1)].

Rectangular Solids from Blocks [09/25/1998]

How many rectangular solids can be made from "n" cube-shaped blocks?

Recurrence Relation for a Pell Equation [11/09/1999]

Can you help me find a recurrence relation for generating solutions to the Pell equation x^2 - 5y^2 = 1?

Relationship Between GCF and LCM [05/22/2002]

What is the exact relationship between the gcf or gcd and the lcm of two numbers?

Relatively Prime [10/07/1999]

What does the term relatively prime mean, and how can you determine if two numbers are relative primes?

Relatively Prime Pythagorean Triples [09/13/1997]

Questions about Pythagorean triples.

Remainders, Pigeons, and Pigeonholes [03/26/2003]

Given 17 integers, prove that it is always possible to select 5 of the 17 whose sum is divisible by 5.

Remainder when Dividing Large Numbers [04/17/2001]

How can I find the remainder when (12371^56 + 34)^28 is divided by 111?

Repeating Decimals [04/28/1999]

I am interested in finding longer repeating groups in number tails of repeating decimals.

Repeating Decimals - Rational or Irrational? [09/11/2001]

Are 0.252252225... and 0.125126127... rational or irrational?

Repeating Digits of Fractions [04/28/1999]

Do you know any theorems relating to the length of the repeating portion of the decimal representation of fractions?

Reversal of Age Digits Every Eleven Years [11/06/2007]

Every 11 years, my age is the exact reverse of my mother's age. When I was 13 she was 31, when I was 24 she was 42, and so on. Why does this work?

Reversed Digits Theorem [06/24/1999]

For a positive integer abc..., if (abc...)^n = xyz... and if (a+b+c+...)^n = x+y+z+..., how can I prove that (...cba)^n = ...zyx?

Reverse Modulus Operator [10/09/2001]

Is there an operator that would return 2 when we we do 6 * 0, * being this new operator?

Roots of ax^f2 bx+c = 0 [05/22/1997]

Prove that if a,b,c are odd integers, then the roots of ax^2 bx+c=0 are not rational.

Rounding Negative Numbers [05/23/2007]

I was wondering what -0.5 is rounded to the nearest integer?

RSA Encryption [04/25/2002]

Decrypt the encrypted message in ciphertext C to find the original plaintext, a string of English letters.

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?

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?

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