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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'?
 Negative Bases [05/13/2002]

How would you represent a number (e.g., 16) in base 2?
 Negative Numbers in Binary [08/19/1999]

How do I represent 53 as a binary number? Is it 110101? Wouldn't that be
+53?
 Negative Numbers to Powers [11/23/2001]

If u is an irrational number and x is a negative number, what is x^u? How
do I determine even whether that number is positive or negative?
 n, e Help Deciphering It [06/21/2012]

An encoded message stumps a friend familiar with Caesar ciphers. Doctor Vogler
introduces the powermod algorithm to crack the case.
 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^22y^2=51.
 Noninteger 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?
 Nonnegative Integers [11/15/2001]

How many nonnegative integers consisting of 13 digits are divisible by
5? How many nonnegative integers consisting of 13 different digits are
divisible by 5?
 NonPeriodic, NonTerminating Decimals [06/26/2001]

Why is a nonperiodic, nonterminating decimal an irrational number?
 Nonterminating Decimal Representations of Fractions [11/10/1997]

Why when you take a finite, limited quantity like onethird and turn it
into a decimal do you get .333... on into infinity?
 Normal Numbers [10/19/1999]

Is there a nonprobabilistic 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, ...?
 NoSolution 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 nonzero 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^p2^p)(7^q2^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^2b 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 email 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?
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