See also the
Dr. Math FAQ:
0.9999 = 1
0 to 0 power
n to 0 power
0! = 1
dividing by 0
Browse College Number Theory
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Selected answers to common questions:
Testing for primality.
- Finding the Two Squares [06/11/2003]
One of Fermat's theorems says that every prime number that yields a
remainder of 1 when divided by 4 can be expressed as the sum of two
integer squares (e.g.: 97 = 4^2 + 9^2). This theorem was proven by
Fermat. What methods are known for determining the two squares?
- Find Remainders: 3^2002/26, 5^2002/26 [09/21/2002]
Find the remainders obtained when 3^2002 and 5^2002 are divided by 26;
show that 3^2002 + 5^2002 is divisible by 26.
- Find the Flaw [08/02/2001]
I don't understand where the following proof goes wrong...
- Find the Smallest Triangle [05/25/2001]
A triangle has sides whose lengths are consecutive integers. Its area is
a multiple of 20. Find the smallest triangle that satisfies these
- Find the Solution: r^2 + s^2 = c. [01/28/2003]
Given c, find a^2 + b^2 = c^2.
- Finite Series and Greatest Integers [03/06/2003]
For n a positive integer, let t(n) denote the number of positive
divisors of n (including n and 1), and let s(n) denote the sum of
these divisors. Prove the following:...
- Formula for phi(n) [05/09/2003]
Find a formula for phi(n) where n is any positive integer.
- 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 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.
- 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?
- 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?
- Genus of a Plane Curve [05/04/2007]
How one can determine the genus of a given curve F(X,Y) in
- Gosper's Verstion of Stirling's Formula [05/30/2002]
I came across a formula by Gosper which seems to be an improvement
on Stirling's formula. Can you show me how to derive this formula?
- 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 Factor [03/28/1997]
How do you find the greatest common factor?
- Greatest Common Factor of Numbers Composed of All Ones [06/30/2008]
A quick proof of why any two numbers composed entirely of ones, with
one number having one more, such as 1111 and 11111, are relatively prime.
- 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.
- Group Sizes and Remainders [05/19/2002]
A farmer can divide his sheep into equal groups of 17; but for any
group size less than 17, he gets a remainder of one less than the
group size. How many sheep does he have?
- Hill Cypher [03/19/2001]
What is the Hill Cypher? Can I decode the Hill Cypher by finding the
inverse of a matrix with all its elements in mod n arithmetic?
- How Many Digits Are in the Root? [07/03/2008]
What is a method for finding the number of digits in the square root
of a 29-digit number? How about any root of any number with a given
number of digits?
- How Many Digits in Graham's Number? [11/11/2005]
I have heard that Graham's number is the largest number with
mathematical use. I have seen it expressed in arrow notation but
that does not give me a sense of how large it is. Is there a way to
express the number of digits it contains?
- Idempotents of Z(n) [10/10/2000]
What are the idempotents of Z(n) when n is twice a prime?
- Inconstructible Regular Polygon [02/22/2002]
I've been trying to find a proof that a regular polygon with n sides is
inconstructible if n is not a Fermat prime number.
- Induction Proof with Inequalities [07/03/2001]
Prove by induction that (1 + x)^n >= (1 + nx), where n is a non- negative
- An Inductive Proof [02/13/2003]
If gcd(n,m) = 1, prove gcd(Rn,Rm) = 1.
- Inductive Proof of Divisibility [06/25/2002]
How do you prove that for any integer n the number (n^5)-n is
divisible by 30?
- Infinite Continued Fraction [05/15/2002]
What can you determine about the value of the infinite continued
- Integer Iteration Function [12/24/2003]
Let X be a positive integer, A be the number of even digits in that
integer, B be the number of odd digits and C be the number of total
digits. We create the new integer ABC and then we apply that process
repeatedly. We will eventually get the number 123! How can we prove
- Integer Logic Puzzle [04/22/2001]
Two integers, m and n, each between 2 and 100 inclusive, have been
chosen. The product is given to mathematician X and the sum to
mathematician Y... find the integers.
- Integer Proof Using Diophantine Equation [01/10/2005]
How do you prove that the integer 26 is the only integer preceded by a
a square (25) and followed by a cube (27)?
- Integer Solutions of ax + by = c [04/03/2001]
Given the equation 5y - 3x = 1, how can I find solution points where x
and y are both integers? Also, how can I show that there will always be
integer points (x,y) in ax + by = c if a, b and c are all integers?
- Integer Solutions to 9r^3 - t^3 - s^3 = 6rst [04/25/2013]
A number theorist exchanges observations with Doctor Vogler, touching on elliptic
curves, Cardano's method for solving cubics, and Fermat's Last Theorem for small odd
- Integer Solutions to a^2 - b^2 = k for a Given Integer k [05/07/2005]
How to find integer solutions (a,b) to the equation a^2 - b^2 = k for
a given integer k.
- Integer Solutions to a Cubic Equation [04/11/2005]
Fermat's method of infinite descent is used to show that the cubic
equation (a^3) + (2b^3) + (4c^3) - 4abc = 0, with a, b, and c whole
numbers and without a=b=c=0, has no solution.
- Interesting Diophantine Equation [12/06/2005]
Find all integers x such that x^2 + 3^x is the square of an integer.
- Interesting Integer Problem with Diophantine Equations [06/21/2005]
Two positive integers are such that the difference of their squares is
a cube and the difference of their cubes is a square. Find the
smallest possible pair and a general solution for all pairs (a,b) that
satisfy the statement.
- An Introduction to Basic Diophantine Equations [08/27/2007]
A birdcage contains both 2-legged and 1-legged birds, and there are a
total of 11 legs in the cage. Use a Diophantine equation to find all
possible combinations of birds.