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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.
 Introducution to Algebraic Numbers and Integers [04/15/2008]

While reading about the classifications of complex numbers, such as
real, irrational, and so on, I came upon a reference to 'algebraic
numbers' and 'algebraic integers'. Can you tell me about those?
 Inverse Function for Natural Numbers [6/10/1996]

I've got a question about the function n = 0.5((a+b)^2+3a+b), which is a
onetoone bijection from pairs (a,b) of natural numbers to single
natural numbers n.
 Irrationality of e+pi and e*pi [09/24/2001]

I have read that it is unknown whether either E+Pi or E*Pi is an
irrational number. How can we prove that at most one of the two numbers
is rational?
 Irrational Numbers x,y, x^y Rational? [09/28/2001]

Are there any irrational numbers x and y such that x^y is rational?
 The Jumping Frog of Coordinate County [11/03/2009]

A frog starts at (0,0) and always jumps a distance of exactly 1 unit
to a point with rational coordinates. Show that it is possible for the
frog to reach the point (1/5,1/17) and not to reach (0,1/4).
 Lagrange's Theorem [02/27/2001]

In your archives you show proofs of Lagrange's theorem that every
positive integer can be expressed as the sum of four squares, but is
there an algorithm for identifying which four squares?
 LaGrange's Theorem [02/24/2001]

Please explain LaGrange's Theorem on the number of roots of a polynomial.
 Large Prime Numbers [12/17/1997]

Is there an algorithm to determine whether a very large number is prime?
 Large Prime Numbers [01/13/2009]

What is the largest prime number less than which all primes are known?
 Largest Mersenne Prime [1/19/1995]

What is the largest currently known Mersenne prime?
 Last Eight Digits of Z [01/18/2002]

Consider the recursive function f(n)=13^(f(n1)), where f(0)=13^13.
A=13^13, B=13^(13^13), etc. A=f(0), B=f(A), C=f(B),... Z=f(Y). What are
the last 8 digits of Z?
 Last Four Digits of the Fibonacci Numbers [05/06/2001]

Show that there is a number ending with four zeros in the Fibonacci
sequence; prove that the Fibonacci sequence has a cycle for the last four
digits with a length of 15,000.
 Linear Congruences of Gaussian Integers [04/11/2003]

When does the linear congruence zx congruent to 1 (mod m), for z, x,
and m all Gaussian integers, have a solution? Also, when do we say
that two Gaussian integers are relatively prime?
 Linear Diophantine Equation [02/05/2003]

Given positive integers a and b such that ab^2, b^2a^3, a^3b^4,
b^4a^5,....., prove that a = b.
 Linear Proof [11/07/2001]

We say that f is linear provided that for every x, y in its domain,
f(x+y) = f(x) + f(y). Show that if f is linear and continuous on R (the
set of real numbers), then f is defined by f(x) = cx for some c belong to
R.
 Linear Recurrance Relations [08/10/2004]

Is there a general approach to taking a pattern that is defined
recursively and finding an explicit definition for it?
 List of Perfect Numbers [3/24/1995]

Do you know any Perfect Numbers besides 6, 28, and 496? Do you know where
I can get a list of Perfect Numbers?
 Mathematical Induction [01/28/2002]

Use Mathemetical Induction to prove that any postage of at least 8 cents
can be obtained using 3 and 5cent stamps.
 Mathematica's PrimitiveRoot[] Function [11/14/2002]

My current project is to develop a fast primitive root finder.
 Maximizing Irregular Polygon Area: Which Circle? [05/01/2011]

How do you determine the radius of the circle that maximizes the area of an irregular
ngon circumscribed on it? With the Pari computer algebra system, Doctor Vogler
approaches the question using numerical techniques such as Newton's Method and a
binary search, which suggests that no closedform expression exists.
 Minimizing the Number of Stamps Needed for Postage [11/20/2007]

A student wants to write a computer program to find the minimum number
of stamps needed to create a desired postage given the various stamp
values that are currently available. Doctor Tom explains why that sort
of problem presents a very difficult challenge.
 Missile Launch Code [08/03/2003]

What kind of information could you give all 10 people such that if any
3 of them were to get together, they would be able to launch the
missiles, but if there were only 2 of them, the information would be
insufficient to figure out the code?
 Mod, Inverses, and Number Theory [11/27/2001]

I already have values for e, p and q, and need to find a value for d in:
d = e^1 mod (p1)(q1).
 Modular Form Ingredients in Fermat's Last Theorem [12/13/2000]

The proof of Fermat's Last Theorem shows how the Lseries of elliptic
curves and the Mseries of modular forms correspond, and how modular
forms are composed of different "ingredients." Can you provide a
description of the ingredients of a modular form?
 Modular Operators and Multiple Variables? [05/28/2011]

A student with six linear equations in two variables wonders if the Euclidean
algorithm would solve it. Doctor Vogler simplifies the system by applying the Extended
Euclidean Algorithm, and introducing the Chinese Remainder Theorem.
 Modular Square Roots [03/24/2006]

Is there an algorithm for finding modular square roots where the
modulus is either a square or an odd composite?
 Modulus Congruence Proof [04/18/2001]

How can I prove 2^(3n+2)+21n = 4 mod (49)?
 Modulus Proof [04/16/2001]

Can you please show me why m^(2^n) = 1 mod(2^(n+2)) when m is an odd
integer?
 More on Order of Operations [02/13/2000]

I have found contradictory information on the precedence of the
multiplication and division operations. Is there a universally accepted
rule for the order of these operations?
 Multiplication of Integers Modulo (2^16 + 1) [10/18/2002]

Prove that 2^16 * 2^15 mod (2^16 + 1) = 2^15 + 1.
 Multiplicative Groups of Order (p1) [05/12/2000]

What is the proof that primitive roots for multiplicative groups of order
(p1), where p is prime, exist? Is there an algorithm for finding them?
 No Solution: y^2 = x^3 + 7 [03/17/2003]

Show that y^2 = x^3 + 7 has no integer solution.
 A Number Digits Puzzle [02/23/2001]

How can I determine all positive integers with the property that they are
one more than the sum of the squares of their digits?
 Numbers in the Fibonacci Sequence [07/19/2001]

How can I show that there is a number in the Fibonacci sequence that ends
in 999999999999 ? For what numbers n is there a number in the Fibonacci
sequence that ends in n of 9 ?
 Number Theory and Modular Arithmetic [01/27/2006]

If the order of an element mod(p) = n, is it true that the order of
this element mod(p^k) = n * p^{k1}, where p is odd prime and k > 1?
 Number Theory  Perfect Square [5/26/1996]

Find all the possible values of n...
 Number Theory: Sum of Cube [7/6/1996]

Generate the sum of cube for a given integer n...
 Odd Numbers and Modulo 8 [09/06/2004]

Is it true that any odd number greater than 1, multiplied by itself,
is congruent to 1 modulo 8? If so, why?
 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.
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