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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'?



Nested Square Roots [07/17/1998]
Solve for n where n = sqrt(6 + sqrt(6 + sqrt6 + ...

Newton's Method and Continued Fractions [10/06/1999]
Can you clarify some points on Newton's method of finding square roots without a calculator, and on the continued fraction algorithm (CFA)?

n Factorial - Prove Lower Bound is n^(n/2) [09/11/2001]
I am trying to prove the following: n^(n/2) <= n!

No Integer Solution [10/21/2001]
I don't know how to prove that the following equation has no solution in Z (integers): z^2-2y^2=51.

Non-integer Powers and Exponents [01/06/1999]
How do you find x^n, where n can be an integer, a fraction, a decimal, or an irrational number?

Non-negative Integers [11/15/2001]
How many nonnegative integers consisting of 1-3 digits are divisible by 5? How many nonnegative integers consisting of 1-3 different digits are divisible by 5?

Non-Periodic, Non-Terminating Decimals [06/26/2001]
Why is a non-periodic, non-terminating decimal an irrational number?

Non-terminating Decimal Representations of Fractions [11/10/1997]
Why when you take a finite, limited quantity like one-third and turn it into a decimal do you get .333... on into infinity?

Normal Numbers [10/19/1999]
Is there a non-probabilistic proof of the existence of normal numbers? Is there an algorithm to produce a number r that is normal in two or more different bases? What about irrational numbers like pi, e, or the square roots of 2,3,5, ...?

No-Solution Equations [11/09/2002]
For any positive integer n, let S(n) denote the sum of its digits. Show that the equation n + S(n) = 1,000,000 has no solution. Then solve the equation n + S(n) = 1,000,000,000.

No Solution: y^2 = x^3 + 7 [03/17/2003]
Show that y^2 = x^3 + 7 has no integer solution.

Number and Its Square Using All 9 Digits Exactly Once [05/22/1998]
Using multiplication facts to find all the whole numbers for which the number and its square together use exactly nine digits 1, 2, 3, ..., 9 only once.

Number Base Convention, Consistency -- and Context [05/08/2012]
A retired engineer proposes a better notational system for representing number bases. Perceiving consistency as his main intent, Doctor Peterson troubleshoots aspects of the proposal while agreeing with some of its underlying motivations.

Number Bases beyond 36 [06/22/2003]
What characters are used to represent numbers in a base greater than 37, i.e. beyond Z?

A Number Digits Puzzle [02/23/2001]
How can I determine all positive integers with the property that they are one more than the sum of the squares of their digits?

Number of Digits in a Factorial [09/18/2005]
Is there a way to determine the number of digits that result from a factorial? For instance, 3! = 6 so there is 1 digit and 4! = 24 so there are 2 digits. But what about something much larger, like 125!?

The Number of Divisors of an Integer [04/02/1998]
Formula and proof for the total number of divisors of any integer.

The Number of Zeros in a Factorial [10/01/1998]
How many zeros come after the last non-zero digit of 20,000,000! ?

Numbers and Digit Sums [12/03/1997]
How many numbers between 0 and 99,999 are there whose digits add up to 20?

Numbers in the Fibonacci Sequence [07/19/2001]
How can I show that there is a number in the Fibonacci sequence that ends in 999999999999 ? For what numbers n is there a number in the Fibonacci sequence that ends in n of 9 ?

Numbers Raised to the Negative Power [11/14/2001]
I know that 5^(-N) = 1/5^N. I would like to know why.

Numbers with 12 Factors.... [11/15/1996]
I have to find two numbers that have exactly 12 factors...

Numbers with the Digit 3 [10/27/1999]
In how many numbers between 1000 and 9999 does the digit 3 occur?

Number Theory - Perfect Square [5/26/1996]
Find all the possible values of n...

Number Theory: Primes [07/10/2003]
Find all primes p,q such that ((7^p-2^p)(7^q-2^q))/(pq) is an integer.

Number Theory Proofs [06/24/1999]
How can I prove that the equations (x,y) = g and xy = b can be solved simultaneously if and only if g^2|b for integers g, b?

Numerically Equal Volumes and Surface Areas [06/04/2001]
Find all rectangular solids with integral dimensions, the volumes and surface areas of which are numerically equal.

Odd Digits of Square Numbers [02/07/1999]
Why are there no square numbers other than 1 and 9 that consist entirely of odd digits?

Odd Numbers and Modulo 8 [09/06/2004]
Is it true that any odd number greater than 1, multiplied by itself, is congruent to 1 modulo 8? If so, why?

Odd Perfect Numbers [01/23/2001]
Is this a proof that there are no perfect numbers?

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.

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