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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.

Congruum Problem [04/04/2002]
I have found a reference to Fibonacci and his congruum problem. But something has me stumped...

Decimal To Fraction Conversion [06/25/1998]
I am trying to find a method (one that can be programmed on a PC) to convert the decimal part of a real number to a fraction represented by integers for the numerator and denominator.

Erdos' Proof of Bertrand's Postulate [3/23/1997]
I am looking for a proof of Bertrand's Postulate (there exists a prime number between n and 2n (n>2)), a.k.a. Chebychev's Theorem.

Fermat's Last Theorem [02/02/2002]
An outline of the proof.

Fermat's Last Theorem - Disproof? [05/09/1998]
I would like to ask you whether there is any problem with the following disproof of Fermat's last theorem...

Making Numbers into Palindromic Numbers [10/11/1995]
Apparently almost any number can be made into a palindromic number by reversing the digits and adding and then repeating the steps until you get a palindromic number. Is there a list that tells which numbers can be made into palindromic sums and how many steps would be required?

Modular Forms and Elliptic Curves: Taniyama-Shimura [10/30/1997]
I watched a PBS show on Fermat's last theorem and they kept talking about modular forms and elliptic curves and how they are related. What are they, and how so they relate to one another?

The Number That Begins With 123454321 [4/3/1995]
Prove there exists a value of n such that 2^(n) 2 to the power n begins in decimal notation 123454321 and the rest of the digits are whatever they want to be.

Palindromic Squares [07/28/1997]
Do you know any numbers besides 14641 where both the number and its square root read the same left to right as right to left?

Prime Number Information [4/15/1995]
Prime numbers fascinate me, but information about them is hard to find....

1 + 2^n + 3^n + 4^n Divisible by 5 [03/22/2003]
Find all positive integers n for which 1 + 2^n + 3^n + 4^n is divisible by 5, and prove that your answer is correct.

Abelian Groups Cyclic [03/05/2002]
Prove that every abelian group of order 6 is cyclic.

Adding Rational and Irrational Numbers [11/07/1999]
How can you prove that a rational number added to an irrational number results in an irrational number?

Algebraic Proof about Product of Twin Primes [04/23/2008]
Show that one more than the product of any two twin primes is always a perfect square.

Algorithm to Print Sums of Inverses [09/29/2004]
Assume it's proven that any number n>32 can be written in the form n = m_1 + m_2 + ... + m_k, such that the sum of the inverses of m_i is 1: 1/m_1 + ... + 1/m_k = 1. How can I find an algorithm that prints all the possible m_i's for a given number n?

Analyzing a Number Mystery [04/10/2006]
Performing a certain process on any random four-digit number always leads to the result of 6174 after some number of iterations. Why?

Antifirst Numbers [10/23/2000]
An antifirst number is a number with more divisors than every number before it. I need to write a program that will calculate all the antifirst numbers between 1 and 2,000,000,000.

Approximating Pi with Continued Fractions [03/18/2006]
Pi is approximated by 22/7. How can you generate increasingly accurate approximations of pi using the division of one integer by another?

Are All Perfect Numbers Even? [01/16/1997]
Has it been proved that perfect numbers must be even?

Arithmetic and Geometric Means [11/05/2000]
How can I use Lagrange multipliers to prove that the geometric mean of three positive numbers is less than or equal to the arithmetic mean?

Arithmetic/Geometric Mean Inequality Theorem [04/15/2001]
Prove the AM-GM (arithmetic mean - geometric mean) inequality theorem (prove that (x1+x2+x3+...+xn)/n is greater than or equal to (x1*x2*x3* ...*xn)^(1/n).

Arithmetic Progression Proof [07/19/2006]
How can you prove that the product of four positive integers in an arithmetic progression cannot be the square of an integer?

Artin's Conjecture [05/31/2001]
Which numbers can never be primitive roots and how do we prove that?

Average Age at a Party [10/27/1999]
How can I find b+g if the average age of b boys is g, and the average age of g girls is b, and the average age of everyone, including the 42-year- old teacher, is b+g?

Balanced Ternary Notation [04/06/2002]
The place values in a base 3 number system are powers of 3. Suppose the digits are 1, 0 and -1. The base 10 number 35 is written as 110-1 in this base 3 system. Write this base 3 notation for the base 10 numbers 1 through 35...

Base 2001 Representation of n! [10/05/2003]
Find all positive integers n such that the base 2001 representation of n! consists entirely of 0's and 1's.

Binary Operations Proof [02/26/2009]
Let s be a finite set with an associative binary operation * (it is not given that the operation has an identity). Prove that there is an element such that a * a = a.

The Birch and Swinnerton-Dyer Conjecture [06/20/2012]
What's the Birch and Swinnerton-Dyer Conjecture? Doctor Vogler outlines how "B and SD" connects the rank of elliptic curves with their Hasse-Weil L-functions.

Bounding Mixed Exponential-Algebraic Diophantine Equations [10/21/2010]
A student knows one solution for Diophantine equations that include a mix of exponential and algebraic terms. To put reasonable upper bounds in the hunt for more, Doctor Vogler outlines applications of Baker's Theorem and lattice reduction.

Calculating Length of Repetend of Reciprocals [10/26/2004]
Is there any way of finding out the number of digits in the period of a reciprocal without doing the long division for 1/n?

Calculating N Factorial [10/26/1999]
Is there a way of calculating n! quickly and exactly without crunching all the intermediate factors?

Carmichael Numbers [10/31/1997]
Why must a Carmichael number be the product of at least three distinct primes? Why is n a Carmichael number iff (p-1) divides (n-1) for every prime p dividing n?

Carmichael's Theorem, Lambda Function [06/11/2003]
Finding the last 8 digits of 9^9^9^9.

Catalan Numbers and Probability [05/29/2003]
Twenty persons want to buy a $10 ticket each. Ten of them have a $10 note and others have a $20 note. The person at the ticket counter has no money to start with. What is the probability that the person at the ticket counter will not have a change problem?

Catalan's Conjecture [06/02/2007]
Find all integer solutions a and b such that a^b = b^a + 1.

Cauchy-Schwarz Inequality [11/13/1999]
How can I prove the Cauchy-Schwarz inequality, i.e. that (sum[j=1 to n](a_j)(b_j))^2 is less than or equal to (sum[j=1 to n](j(a_j)^2))* (sum[j=1 to n](((b_j)^2)/j))?

Challenging Diophantine Equation [09/30/2004]
Find all solutions of 2^m = 3^n + 5. The case bothering me is when I have shown m has to be of form 4k+1 and n of the form 4K+3 for some k,K and that K has to be even to ensure 3^(4K+3)+5 is congruent to 0 mod 32.

Coefficients in a Trinomial Expansion [04/24/2001]
In the expansion of (a+b+c)^6, what is the coefficient of a^2b^2c^2?

Combinatorics: Ramsey Theory [01/12/1998]
Could you help me with a detailed explanation of the theory and a concrete example?

Compressing Numbers and the Shannon Limit [05/12/2004]
Is it possible to find a formula that takes X (a large number) and creates another expression which equals X but is shorter in length than X? For example, take 390,625 which uses six characters. I need a formula which calculates something like 5^8, which equals 390,625 but is only three characters instead of six.

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