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
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
Infinite number of primes?
Testing for primality.
What is 'mod'?
- Formula for Connection between Rows of Pascal's Triangle [11/15/2003]
Find a formula connecting any (k+1) coefficients in the nth row of the
Pascal Triangle with a single coefficient in the (n+k)th row.
- Formula for Counting Triangles [03/16/2000]
How many equilateral triangles of integer-length sides are in an
equilateral triangle n units on a side?
- Formula for Factors of a Number [11/3/1996]
How many triangles can you draw on a square grid of dots of size x*x?
- Formula for Pythagorean Triples [10/23/1997]
Is this formula: a = (m^2-n^2); b = 2mn; c = (m^2+n^2) correct for all
- Formula for Sums of Counting Numbers [09/19/2002]
Use the formula for the sum of the first n counting numbers to find 9+
- Formula for the Extraction of a Digit [06/18/2001]
Is there a formula to extract any digit of a given number?
- Formula for the First Day of a Year [03/18/1998]
Is there an equation to find the first day of a year given the year?
- Formulas for Primes [09/09/2002]
Prove that n^2 + n + 41 does not always produce a prime number for any
whole number n. Explain why n^2 + 8n + 15 never produces a prime
- Four-Digit Palindromes [10/21/1998]
Why is every four-digit palindrome divisible by 11?
- Four-digit Palindromes Divisible by 11 [02/10/1999]
Why are four-digit palindromes divisible by 11?
- Four Positive Integers, Any 3 Sum to a Square [10/06/2002]
Find four distinct positive integers, a, b, c, and d, such that each
of the four sums a+b+c, a+b+d, a+c+d, and b+c+d is the square of an
integer. Show that infinitely many quadruples (a,b,c,d) with this
property can be created.
- Four Variable Diophantine Expression [05/10/2008]
For what pairs of different positive integers is the value a/(a+1) +
b/(b+1) + c/(c+1) + d/(d+1) an integer? How would I solve it?
- Fraction Algorithm [03/19/2002]
I have been having trouble making an application that can convert a
finite decimal to a fraction without doing 78349/1000000.
- Fractions between 0 and 1 [07/29/2001]
Is there a way to find the number of different (no equivalent fractions)
fractions between 0 and 1 with denominators from 2 to 100 without writing
out every fraction and counting them?
- Frequency of Digits in Pi [04/05/2001]
What digit occurs least frequently in pi?
- From 0, 1 to Base 10: The View from Inside a Calculator [11/25/2016]
A teen wonders how digital machines output in base ten despite using only zeros
and ones for the actual number-crunching. With an example of division in base-two and
the more familiar decimal system, Doctor Peterson puts his finger on the key insight.
- From Reduction to Induction [02/03/2011]
Replace any two numbers x and y from (1, 2, ..., n) with the new single quantity x + y
+ xy; continue in this way until only one number remains. To find a formula for the
smallest number possible from this procedure, Doctor Jacques lays the groundwork
for a proof by induction.
- Fundamental Theorem of Algebra [01/25/2001]
What exactly is the Fundamental Theorem of Algebra?
- Fundamental Theorem of Arithmetic [07/08/1997]
What's so fundamental about the fundamental theorem of arithmetic?
- Fundamental Theorem of Arithmetic [10/23/2001]
How do I prove that the cube root of 2 is an irrational number using the
Fundamental Theorem of Arithmetic?
- Fundamental Theorems [10/02/2000]
What are the fundamental theorems of algebra and arithmetic?
- Gaussian Integers [05/12/1999]
Are all real prime numbers also Gaussian prime numbers?
- GCD Even/Odd Proof [10/26/2001]
If m is greater than nn and a,m,n are positive with m not equal to n,
prove that the GCD of (a^2^m+1, a^2^n+1) = 1 if a is even; and 2 if a is
- General Formula to Find Prime Numbers? [10/12/2005]
I was wondering if it is possible that there exists a general formula to know what numbers are prime, or has it been proven that no such formula could exist? What evidence do we have for either case?
- Generalised 'Fibonacci' Series and Phi [02/10/2002]
A Fibonacci-style series that starts with any two numbers and adds
successive items produces a ratio of successive items that converges to
phi in about the same number of terms as for the basic Fibonacci series.
Is this well known and provable?
- Generalized Definition of Prime Numbers [02/01/2001]
Examining an extended definition of a prime number.
- Generalized Gauss Sum [07/18/2003]
Is there a formula that computes the sum of numbers from m to n, where
m and n could be positive, negative, or zero?
- General Observation on Prime Numbers [09/03/2004]
Is it true that all prime numbers greater than 5 are of the form 6n +
1 or 6n - 1? I read this on a website, but it's hard to believe.
- Generating Function of Catalan Numbers [04/04/2000]
Can you explain the recurrence relation for the Catalan numbers?
- Generating Pythagorean Triples [03/17/2001]
I need to generate the sixteen primitive Pythagorean triples, and to find
how many there are such that the numbers of the triplet lie between 1 and
- Getting 0.99999... [04/15/1998]
Is there any mathematical way to get 0.99999999999......?
- Given Irrational Numbers a,b, Is a^b Rational? [09/26/2001]
Is it possible to demonstrate that there are irrational numbers a,b such
that a^b is a rational number?
- The Golden Ratio [02/23/1998]
I know that the limit of the ratios of the Fibonnaci sequence is the
golden mean, but I would like to see a proof.
- Graphing y = (-2)^x [05/27/2005]
How do I graph y = (-2)^x, or any function where you have a negative
number raised to a power?
- Graph of y = (-n)^x [01/17/2005]
I am curious as to what the graph of y = (-n)^(x) would look like,
such as y = (-2)^x. My graphing calculator will not show the graph as
anything, but displays many real values in the table of values.
- Graphs - Proving the Infinite Ramsey Theory [11/10/1997]
In a graph with infinite "points," if we colour the lines with two colors
we'll have either a red or a blue infinite chain of lines, an infinite
number of points, all of them joined to each other with the same
- Greatest Common Divisor of Factorials [02/20/2015]
An adult struggles to prove that x is the greatest common divisor of two particular
factorial expressions if and only if it is prime. Recognizing Wilson's Theorem, Doctor
Ali walks through the proof.
- Greatest Common Factor [03/28/1997]
How do you find the greatest common factor?
- Greatest Impossible Score in a Game [01/26/2003]
If the two values possible in a game are p and q, the greatest
impossible score is (pq - p - q). Why is it that?
- Greatest Integer Equation [08/06/2003]
I am trying to correctly interpret [[x]]^2 + [[y]]^2 = 1, where f(x)=
[[x]], is the Greatest Integer function.