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Browse College Number Theory
Stars indicate particularly interesting answers or
good places to begin browsing.
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
conditions.
 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
number.
 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
Z[X,Y]?
 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
colour...
 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 29digit 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
integer.
 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
fraction [1;1,2,3,1,2,3,1,2,3....]?
 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
that?
 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
powers.
 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 2legged and 1legged birds, and there are a
total of 11 legs in the cage. Use a Diophantine equation to find all
possible combinations of birds.
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