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Browse College Number Theory
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Selected answers to common questions:
    Testing for primality.

The Official Euclidean Algorithm [11/16/2000]
Can you state briefly the "official" Euclidean Algorithm?

One Proof that a Diophantine Equation in Two Variables Has Only Three Solutions [01/30/2011]
Having found some solutions of a Diophantine equation with variable exponents, a student seeks all solutions -- and proof thereof. Invoking modular arithmetic, Pythagorean triples, and coprimality, Doctor Vogler demonstrates how he would do it.

Only Two Abelian Groups [02/25/2003]
Show that any group with order p^2, p is a prime, is Abelian. Show that up to isomorphism that only two such groups exist.

Orbits of the Baker's Map Function [12/11/2000]
I'm investigating the number of orbits a particular input has for the Baker's map function... why are all the values with 1 orbit the primes of primitive root 2?

Overview of Riemann's Zeta Function and Prime Numbers [04/11/2006]
Can you please give an overview of the importance of the Zeta function and finding prime numbers? Why is the Zeta function such a hot topic in the field of looking for prime numbers?

Pairs of Integers [08/16/1997]
Show that there are infinitely many pairs of integers(x,y) such that x|y**2+m and y|x**2+m where m is any chosen integer; moreover gcd(x,y)=1.

Palindrome Problem [2/6/1995]
According to the "Reversing Number Algorithm," it's thought that most (perhaps all) numbers eventually wind up being palindromes. Have you seen any references about this problem?

Partitions and Products [01/02/2003]
What is the best way of dividing an integer into parts so that the product of the parts will be as large as possible? Is there a universal law that tells us what partition will produce the maximum product for any given number? And can such a law be proved?

Patterns in Repeating Decimals [08/06/2003]
Why do certain number sequences repeat in the decimal expansions of common fractions?

Pell-like Diophantine Equations [10/05/2010]
How do you solve exponential Diophantine equations in two variables? Even after seeing recurrence relations and modular arithmetic simplify his original problem, a student remains curious about additional strategies, so Doctor Vogler obliges by outlining more advanced methods.

Perfect Numbers [06/15/2002]
Please show that any even perfect number ends in 6 or 8.

Perfect Square [10/26/2001]
If a and b are positive integers such that (1+ab) divides (a^2+b^2), show that the integer (a^2+b^2)/(1+ab) must be a perfect square.

Perfect Square Proof [10/07/2016]
A teen seeks to prove that only two possible values make a perfect square of a^2 + 3ab + b^2. By factoring, analyzing cases, and adding in some judicious terms, Doctor Ali works through the puzzle.

Pigeonholes and Remainders [12/27/2003]
How can I show that in any collection of 52 distinct positive integers, there will be two distinct numbers whose sum or difference will be divisible by 100?

Polya-Burnside Lemma [07/02/1999]
Given unlimited amounts of 2 types of beads, how many unique necklaces can you make using exactly k beads?

Polynomial Diophantines: Independent Study for Integer Solutions [11/30/2011]
A student seeks suggestions to further his study of polynomial Diophantine equations. Doctor Vogler provides an overview of the big picture, along with several textbook recommendations.

Polynomial Divisible by 7 [11/14/2001]
Prove that 2^(3n+1) + 4^(3n+1) + 1 is divisible by 7.

Power from Its Base and a Little String? [04/11/2012]
Given a base raised to some power, and the last few digits of its decimal expansion after that exponentiation, a student seeks a method for determining the exponent. Doctor Vogler obliges with two examples, relying on a structural property of the product of cyclic groups.

Powers of 2 Proof [03/24/2003]
Prove that any number that is not a power of 2 can be expressed as a sum of two or more consecutive positive integers, but that this is not possible for powers of 2.

Predicting Number of Decimal Digits in a Binary Number [03/10/2004]
I am trying to write a function that will return the number of decimal digits from a binary number without actually converting from binary to decimal. Is there a way to determine which length is correct without going through a complete conversion?

Primality Testing [12/02/2004]
How can I determine if a given number is prime?

Prime Factors, Modular Arithmetic, and Using Pari [08/08/2007]
Suppose we have two positive integers 'a' and 'b'. Is there a method to find a positive integer 'k', such that a + bk = x^2 for some integer 'x'? In other words, how can we find a positive integer 'k', such that 'a + bk' is a square?

Prime Numbers between N and 1+N! [04/25/2003]
For each natural number N, there exists a prime number Q such that N is less than or equal to Q and Q is less than or equal to 1 + N! Two proofs: one that invokes Bertrand's Postulate, the other by contradiction.

Prime Numbers Proof [05/23/2005]
Prove that every prime number is the leg of exactly one right triangle with integer sides.

The Prime Number Theorem [05/26/2008]
Let pi(x) be the number of primes below x. In a book I read, two formulas are given with identical right sides but the two left sides are pi(x) and pi(x) - pi(sqrt(x)) + 1. How can those two expressions be the same so that both formulas are true?

Prime Proofs [10/08/2002]
If a^(n) - 1 is prime, show that a=2 and that n is a prime. If a^(n) + 1 is a prime, show that a is even and that n is a power of 2.

Primes and Squares [05/03/2001]
For what values of prime number p is (2^(p-1)-1)/p a perfect square?

Primes Containing but Not Ending in 123456789 [02/26/2003]
Are there infinitely many primes that contain but do not end in the block of digits 123456789 ?

Primes of the Form 4n+3 [11/07/1999]
Prove that there are infinitely many primes of the form 4n+3 where n is an element of the natural numbers.

Primes: p+1 a Multiple of 6? [10/06/2002]
Prove that if p and p+2 are both prime, then p+1 is divisible by 6. Completely stuck on what to do.

Primitive Elements vs. Generators [05/24/2002]
Prove that x is a primitive element modulo 97 where x is not congruent to 0 if and only if x^32 and x^48 are not congruent to 1 (mod 97).

Primitive Pythagorean Triple Congruences [04/17/2003]
If x,y,z are primitive Pythagorean triples, prove that x+y and x-y are congruent modulo 8 to either 1 or 7.

Primitive Roots of Primes [06/05/2005]
I define the Primitive Roots Index (PRI) of a prime p as the [sum of all its p.r.ís] mod p. A surprising property of PRI which I stumbled upon is its value is always 0, + 1, - 1 for all primes. Is it possible to prove this property for all primes? Is it possible to derive the PRI as a function of the value of p?

Primitive Triples [04/24/2003]
In a primitive Pythagorean triple (x, y, z), why must x+y and x-y be congruent to either 1 or 7 mod 8 ?

Primorials [10/15/2003]
We know that p_1 * p_2 * ... * p_n + 1 is either prime or divisible by a prime not included in the list. But is the second condition necessary? Is the result ever not prime?

Probability of a Sum [08/16/2002]
If the integers m and n are chosen at random between 1 and 100, what is the probability that a number of the form 7^m + 7^n is divisible by 5?

Problem Posed by Fermat [05/04/2001]
Find a right triangle such that the hypotenuse is a square and the sum of the two perpendiculars, or indeed of all three sides, is also a square...

Product of Two Primes [10/27/1999]
How many positive integers less than 100 can be written as the product of the first power of two different primes?

Programs to Find Prime Numbers [11/27/1996]
Can a program be written in BASIC to compute the number of prime numbers smaller than n?

Proof by Induction [05/24/2002]
Prove by induction that (n^7 - n) is divisible by 42.

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