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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'?



Prime Number Proof: p_2n Greater Than 2*p_n [10/07/2001]
Prove that for n greater than 1, p_2n is greater than 2*p_n, where p_n is the nth prime number.

Prime Numbers [07/10/1998]
If you multiply all the prime numbers up to N together, the limit appears to be exp(N) as N gets large. Is there a simple reason for this?

Prime Numbers [12/15/2000]
How can I prove that if A = (P1*P2+1)^4 - 1, where P1 and P2 are two distinct primes; then A is a multiple of three distinct primes?

Prime Numbers and n^2-n+41 [07/23/2003]
How does n^2-n+41 work to produce a prime number for every integer value, and why does it it fail when n = 41 ?

Prime Numbers as the Difference of Two Squares [08/07/2002]
Express the prime numbers 7, 15, and 261 as the difference of two squares.

Prime Numbers between 1 and 150 [7/30/1996]
How many prime numbers are there between 1 and 150?

Prime Numbers in Cryptography [08/14/1999]
What are some practical uses of prime numbers?

Prime Numbers in Different Bases [10/07/1998]
Are all prime numbers the same in all bases? If 21 is a prime, are 10101 (in binary), and 15 (in hexadecimal) also primes?

Prime Number Tests [11/12/1998]
Is the number 55409243 prime? How can you test to see whether a number is prime?

Prime Number Theorems [01/03/1999]
Can you explain the prime number theorem, Mersenne primes, the Lucas- Lehmer test, and the Riemann Hypothesis?

Prime Proofs [10/08/2002]
If a^(n) - 1 is prime, show that a=2 and that n is a prime. If a^(n) + 1 is a prime, show that a is even and that n is a power of 2.

Primes and Repeating Unit Numbers [12/09/1998]
How do you prove this statement: For every prime number there exists a repeated unit number that is a multiple of that prime.

Primes and Squares [05/03/2001]
For what values of prime number p is (2^(p-1)-1)/p a perfect square?

Primes Containing but Not Ending in 123456789 [02/26/2003]
Are there infinitely many primes that contain but do not end in the block of digits 123456789 ?

Primes Greater Than/Less Than Multiples of Six [01/18/2002]
Has the postulate stating that every prime number is either one more or one less than a multiple of six, excluding 2 and 3, been proven?

Primes in the Form n^2 + 1 [04/06/2003]
Let n be a positive integer with n not equal to 1. Prove that if n^2 + 1 is a prime, then n^2 + 1 is expressible in the form 4k + 1 with k in the integers.

Primes of the Form 4n+3 [11/07/1999]
Prove that there are infinitely many primes of the form 4n+3 where n is an element of the natural numbers.

Primes: p+1 a Multiple of 6? [10/06/2002]
Prove that if p and p+2 are both prime, then p+1 is divisible by 6. Completely stuck on what to do.

Primes that are Sums of Primes [06/22/1999]
Is there an nth prime number, p, (other than 5, 17 and 41) that is equal to the sum of the prime numbers up to n? For example, the 7th prime is 17=2+3+5+7.

Primes That Are the Sum of 2 Squares [09/17/1999]
How can I prove that every prime of the form 4m + 1 can be expressed as a sum of two squares?

Prime Triplet [12/07/2001]
The consecutive odd numbers 3,5,7 are all primes. Are there infinitely many such 'prime triplets'?

Primitive Elements vs. Generators [05/24/2002]
Prove that x is a primitive element modulo 97 where x is not congruent to 0 if and only if x^32 and x^48 are not congruent to 1 (mod 97).

Primitive Pythagorean Triples [02/23/1998]
Given a triple of numbers (a, b, c) so that a, b, and c have no common factors and satisfy a^2+b^2 = c^2, make a guess about when a, b, or c is a multiple of 5.

Primorials [10/15/2003]
We know that p_1 * p_2 * ... * p_n + 1 is either prime or divisible by a prime not included in the list. But is the second condition necessary? Is the result ever not prime?

Probability of Divisibility [06/18/2002]
What is the probability that a randomly selected three-digit number is divisible by 5?

Probability of Random Numbers Being Coprime [08/12/1997]
I have heard that the probability of two randomly selected integers being coprime is 6/(pi^2). How do you show this is true?

Problem Posed by Fermat [05/04/2001]
Find a right triangle such that the hypotenuse is a square and the sum of the two perpendiculars, or indeed of all three sides, is also a square...

Product Always an Even Number? [03/17/2002]
The letters a1, a2, a3, a4, a5, a6, a7 represent seven positive whole numbers; b1, b2, b3, b4, b5, b6, b7 represent the same numbers but in a different order. Will the value of the product (a1-b1)(a2-b2)(a3- b3)(a4- b4)(a5-b5)(a6-b6)(a7-b7) always be an even number?

Product and Sum of Digits = Number [10/24/2001]
How many two-digit numbers exist such that when the products of their digits are added to the sums of their digits, the result is equal to the original two-digit number?

Product of Primes [02/27/2002]
Can you provide me with the proof that every non-zero positive integer can be written as a product of primes?

Product of Two Primes [10/27/1999]
How many positive integers less than 100 can be written as the product of the first power of two different primes?

Products of Integers (Even or Odd) [03/23/2002]
How can I prove that the product of two even integers is an even integer and the product of two odd integers is an odd integer?

Programs to Find Prime Numbers [11/27/1996]
Can a program be written in BASIC to compute the number of prime numbers smaller than n?

Program to Convert Number Bases [07/12/1999]
Is there an easier method for converting bases than dividing and collecting the remainders? I want to write a computer program to do this.

Proof by Contraposition [03/06/2002]
How can I prove that n^6 + 2n^5 - n^2 - 2n is divisible by 120?

Proof by Induction [05/24/2002]
Prove by induction that (n^7 - n) is divisible by 42.

Proof by Mathematical Induction [09/24/1999]
Prove the following statement by mathematical induction: for any integer n greater than or equal to 1, x^n - y^n is divisible by x-y where x and y are any integers with x not equal to y.

Proof Involving Legendre Symbol [02/03/2003]
If p, q are both prime odd numbers such that they are not factors of a, and p=q(mod 4a), prove that (a/p)=(a/q).

Proof Involving mod 5 [10/27/2002]
Prove n^2 mod 5 = 1 or 4 when n is an integer not divisible by 5.

Proof of Lagrange's Theorem [11/23/2000]
I am looking for a proof of Lagrange's Theorem, which states that any positive integer can be expressed as the sum of 4 square numbers.

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