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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.
 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
xy**2+m and yx**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?
 Pelllike 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?
 PolyaBurnside 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^(p1)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 xy
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.’s] 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 xy 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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