See also the
Dr. Math FAQ:
0.9999 = 1
0 to 0 power
n to 0 power
0! = 1
dividing by 0
Browse High School Number Theory
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Selected answers to common questions:
Infinite number of primes?
Testing for primality.
What is 'mod'?
- 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^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
- 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
- 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
- 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