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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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Proof that Sqrt(3) is Irrational [08/14/1997]

How does one prove that sqrt(3) is irrational? or others? Is there a general algorithm? How about just for primes?

Proof That the Cube Root of 3 is Irrational [05/22/2000]

How can I show that the cube root of 3 is irrational?

Proof with Pigeonhole Principle [09/20/2001]

Prove that among five points selected inside an equilateral triangle with sides of length 2, there always exists a pair at a distance not greater than 1.

Proof with Powers of 2 and a Product [10/22/2008]

Prove that starting with any power of 2 there is a number such that the product of those two numbers will contain only the digits 1 and 2. For example, 4*3 = 12, 8 * 14 = 112, 64*33 = 2112.

Properties of the Phi Function [01/19/1999]

What are some properties of the phi function? What about the phi function and prime numbers?

Prove 101 the Only Prime [12/14/2002]

In this sequence of integers, all in base 10: 101, 10101, 1010101, 101010101, 10101010101, .......,,,, etc., prove that 101 is the only prime in the sequence.

Prove a and b are Perfect Squares [12/28/2001]

Let a and b be positive integers such that (a,b) = 1 and ab is a perfect square. Prove that a and b are perfect squares.

Prove a = b = c [01/27/2002]

When a^2 + b^2 + c^2 = ab + bc + ca and abc does not equal 0, prove that a = b = c.

Prove that 7 + 17*sqrt(17) is Irrational [09/22/2002]

I need to prove that 7 + 17*sqrt(17) is irrational. I know to set it equal to a/b, and have already proven that any square root of a prime number is irrational.

Prove That an Expression is a Multiple of 10 [12/19/2002]

If a and b are positive integers, prove that (a^5)*(b) - (a)*(b^5) is a multiple of 10.

Prove that Log A is Irrational [06/14/1998]

Can you help me prove that a common log of a number (not powers of 10) is irrational?

Prove that n^3 + 2n is Divisible by 3 [04/15/2001]

Prove by mathematical induction that n^3 + 2n is divisible by 3 for all positive integers.

Prove Twin Primes Greater Than 3 Divisible by 12 [10/08/2002]

Prove that if p and q are twin primes, each greater than 3, then p+q is divisible by 12.

Prove x^2+y^2 Not Divisible by 4 [09/20/2001]

Prove that if x and y are odd, then x^2 + y^2 is even but not divisible by 4.

Proving a Number is Prime [10/13/2004]

How do you really prove that 2 or some other number is a prime number?

Proving a^x = a^y iff x = y [12/13/2000]

How can I prove that a^x = a^y iff y = x for all real numbers x and y?

Proving De Moivre's Theorem [12/03/1997]

Prove De Moivres theorem: - (cos(x)+isin(x))^n = cos(nx) + isin(nx) .

Proving Divisibility [04/30/2002]

Prove that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 15.

Proving Divisibility [09/11/2003]

Prove that (n^2 - n) is divisible by 2 for every integer n; that (n^3 - n) is divisible by 6; and that n^5 - n is divisible by 30.

Proving e is Irrational [11/19/1997]

My professor suggested using a proof by contradiction, but I don't understand how to do it.

Proving Fermat's Last Theorem for N = 4 [05/18/2000]

How can you prove Fermat's Last Theorem for the specific case n = 4?

Proving O(n) [01/23/2001]

How would you prove that an equation is of order n, or n squared?

Proving Perfect Squares [07/05/1998]

Suppose a, b, and c are positive integers, with no factor in common, where 1/a + 1/b = 1/c. Prove that a+b, a-c, and b-c are all perfect squares.

Proving Phi(m) Is Even [04/22/1998]

Explain why phi(m) is always even for m greater than 2...

Proving the Associative Property [02/24/2001]

How can I prove that a binary operation is associative, if all I am given is a table for the operation?

Proving the Properties of Natural Numbers [03/08/2000]

How can you prove or derive the commutative, associative, and distributive properties of numbers?

Proving the Square Root of 2 is Irrational [02/04/2004]

How can you prove that the square root of 2 is irrational using the Rational Root Theorem?

Proving the Square Root of a Prime is Irrational [07/15/1998]

How do you prove that if p is prime, the square root of p is irrational?

Public Key Encryption [03/29/1999]

Examples and discussion of operations used for encryption, including mod.

Pythagorean Quadruplets [12/28/1998]

I am trying to find a formula that generates Pythagorean quadruplets a,b,c,d such that a^2 + b^2 + c^2 = d^2.

Pythagorean Theorem, Fermat's Last Theorem [5/16/1996]

Can the Pythagorean theorem be done with 3 different numbers?

Pythagorean Triple [8/28/1996]

What is the formula for finding the three lengths in a Pythagorean triple where the shortest side is even?

Pythagorean Triples [10/07/1997]

What is a Pythagorean triple?

Pythagorean Triples [04/14/1997]

Why can't all the numbers in a Pythagorean triple be prime?

Pythagorean Triples [07/14/1997]

Is there a formula to determine the solutions to the following equations? a^2 + b^2 = c^2, a^3 + b^3 + c^3 = d^3...

Pythagorean Triples [11/19/1997]

I need to know the first five Pythagorean triples after 3,4,5...

Pythagorean Triples [05/22/1999]

What is the general formula for all sides of any triple?

Pythagorean Triples [05/31/1999]

Is there a procedure for finding Pythagorean triples?

Pythagorean Triples [5/18/1995]

How can the relation between Pythagorean triples be expressed as a formula?

Pythagorean Triples Divisible by 5 [11/17/2000]

Do all right triangles with integer side lengths have a side with a length divisible by 5?

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