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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'?

p, p+8, p+22 Not Prime [10/16/2001]
Prove that there is no positive integer p such that each of the numbers: p, p+8, p+22 is prime.

Prefix for 10^30 Bytes [05/25/2000]
What do you call 1,000,000,000,000,000,000,000,000,000,000 or 10^30 bytes?

Primality Test [11/26/2001]
I want to write a program using Pascal that will verify whether a number is prime.

Primality Testing [04/22/1998]
Is there any fomula to find if a number is a prime?

Primality Testing [12/02/2004]
How can I determine if a given number is prime?

Prime and Consecutive Numbers [11/16/2001]
Why are 3, 5, and 7 the only numbers that appear to be prime and consecutive?

Prime Factor [02/18/2002]
I need to prove that each integer of the form 3n + 2 has a prime factor of this form.

Prime Factors, Modular Arithmetic, and Using Pari [08/08/2007]
Suppose we have two positive integers 'a' and 'b'. Is there a method to find a positive integer 'k', such that a + bk = x^2 for some integer 'x'? In other words, how can we find a positive integer 'k', such that 'a + bk' is a square?

Prime Factors of 4,194,305 [09/20/1999]
How can I find the prime factors of 2^22+1?

Prime Integer Proof [03/24/2002]
Prove that if p is a prime number greater than or equal to 5, then there exists an integer k such that p=sqrt(24k+1).

Prime Number 2001, Sieve of Eratosthenes [01/25/1997]
Is 2001 a prime number?

Prime Number Formula [11/11/2001]
What formula did Leonhard Euler use to find prime numbers?

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?

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