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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'?

Odd Primes and Primitive Roots [11/14/2001]
Given distinct odd primes P and Q, prove that N = PQ has no primitive roots.

The Official Euclidean Algorithm [11/16/2000]
Can you state briefly the "official" Euclidean Algorithm?

One equals Two [07/25/2001]
There is an algebraic manipulation involving division by zero that results in one equals two, or some other contradiction. What is it?

One Plus One isn't Two [1/10/1995]
I was once shown that 1+1 isn't 2, and I don't remember how it was done. Could you please e-mail me with an answer?

One to the Power of Infinity [07/03/2001]
An instructor says that one to the infinite power does not equal one. If this is true, is there a relatively simple explanation?

Operations in Nondecimal Bases [10/16/1999]
How can you subtract, multiply, and divide numbers in other bases?

Ordering Real Numbers [12/11/2000]
Can you describe how to order real numbers?

The Origin of Lucas Numbers [10/08/1998]
I need help with Lucas Numbers - how and why they were created.

Paint Formulas in Base 48 [08/20/1998]
I work for a paint store where our formulas are based on an ounce being 48 parts...

Pairs of Integers [08/16/1997]
Show that there are infinitely many pairs of integers(x,y) such that x|y**2+m and y|x**2+m where m is any chosen integer; moreover gcd(x,y)=1.

Paradox in the Zero Power [09/26/2000]
Why does the product of multiplying a number by itself zero times equal 1?

Partitioning an Integer [11/14/1998]
How many different ways are there of making a number by adding different combinations of three numbers?

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.

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