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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'?



Partitions and Products [01/02/2003]
What is the best way of dividing an integer into parts so that the product of the parts will be as large as possible? Is there a universal law that tells us what partition will produce the maximum product for any given number? And can such a law be proved?

Pascal's Triangle and Fibonacci Formula [02/23/2002]
Prove that the diagonals of Pascal's triangle are the Fibonacci numbers.

Paths to Triangle Points [05/26/1999]
How can I find the number of paths to a point using Pascal's triangle?

Pattern in Period [01/31/1999]
Is there any pattern in odd periods with 1 as numerator?

Patterns in Repeating Decimals [08/06/2003]
Why do certain number sequences repeat in the decimal expansions of common fractions?

Patterns in Rolling 3 Dice [09/16/1997]
I have come out with 216 outcomes when rolling 3 dice; while I was listing all the outcomes, I began to see a pattern...

Pell Pairs of Positive Integer Solutions [03/29/2012]
Solving a Pell equation confuses a student. Doctor Vogler applies modular arithmetic to show the way, then concludes with a survey of online resources into Diophantine equations.

Percentage of Numbers Divisible by 6, 8 [03/18/2003]
What percentage of numbers is divisible by 6? by 8?

Percentages of Prime Numbers [01/31/1999]
Does the percentage of prime numbers at every power of 10 decrease until it reaches a constant value?

Perfect Logarithms [06/24/2002]
What can you tell me about the equation log(abc)= log(a+b+c)?

Perfect Number [7/21/1996]
How can I find a perfect number? What are some reference books about number theory?

Perfect Number Algorithms [03/31/1998]
What is the formula for a computer program that tests whether an integer is a perfect integer?

Perfect Numbers [11/10/1997]
Do the sums of the digits of perfect numbers always equal 1?

Perfect Numbers [06/15/2002]
Please show that any even perfect number ends in 6 or 8.

Perfect Square [10/26/2001]
If a and b are positive integers such that (1+ab) divides (a^2+b^2), show that the integer (a^2+b^2)/(1+ab) must be a perfect square.

Perfect Square [07/23/2001]
If g.c.d.(x, 3) = 1 and g.c.d.(y, 3) = 1, show that x^2 + y^2 cannot be a perfect square.

Perfect Square? [02/18/2002]
If we use the digits 1,2,3,4,5,6,7 each only once to form a 7-digit number, can the resulting number be a perfect square?

Perfect Square, Cube, Fourth Power [01/25/2002]
Find the least integer greater than 1 that is a perfect square, a perfect cube, and a perfect fourth power.

Perfect Square Equation [02/22/2002]
Prove that if n is greater than 1, then nC2 + (n-1)C2 is a perfect square.

Perfect Squares and Irrational Numbers [02/13/2002]
Isn't any non-perfect square an irrational number? What is the number 0.49? Its square root is 0.7, which is neither irrational nor an integer.

Perfect Squares: n+125 and n+201 [01/21/2002]
Find the smallest positive integer n so that n+125 and n+201 are both perfect squares.

Perfect Squares with Congruences [02/16/2002]
Prove that there is no perfect square a^2 whose last digits are 35.

Perimeter of Pascal's Triangle [06/05/2001]
Is there a general formula for finding the perimeter of Pascal's triangle using the number of rows?

The Phi Function [11/21/1998]
What are the conditions on n,m so that phi(n*m) = phi(n)*phi(m)? What is phi(p^n*q^m)?

Polynomial Divisible by 7 [11/14/2001]
Prove that 2^(3n+1) + 4^(3n+1) + 1 is divisible by 7.

A Polynomial in Three Variables with Few Integer Solutions [03/12/2011]
A student seeks proof that a polynomial in n and two other variables has no integer solutions. After a little insight from modular arithmetic and a lot of searching with a computer algebra system, Doctor Vogler turns up many solutions.

Positives and Negatives with Infinity and Zero [10/05/1999]
Are there such things in math as +0, -0, and unsigned 0; and +infinity, - infinity, and unsigned infinity? Are these different?

Powers of 2 Proof [03/24/2003]
Prove that any number that is not a power of 2 can be expressed as a sum of two or more consecutive positive integers, but that this is not possible for powers of 2.

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?

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