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

Congruence Class of 10^n Modulo 11 [04/14/2003]
What is the congruence class of 10^n modulo 11? Use this to determine the remainder when 654321 is divided by 11.

Conjectures vs. Hypotheses [01/12/1999]
What is the difference between the terms 'conjecture' and 'hypothesis'? Should the Riemann hypothesis be the Riemann conjecture?

Consecutive Composite Numbers [07/05/2002]
Find 1000 consecutive composite numbers.

Consecutive Composite Numbers [06/07/2005]
Is it possible to find m consecutive composite numbers where m is any natural number?

Consecutive Integer Proof [09/16/2004]
Is it possible to prove that the product of five consecutive integers cannot be a perfect square?

Counting Infinities [04/29/1997]
I think I have found a flaw in the uncountability proof. Is this possible?

Counting Unique Rational Numbers [11/16/2008]
How many unique simple forms of rational numbers are there of the form p/q, where p and q are non-zero whole numbers less than or equal to n? For example, 1/2 and 2/4 have the same simple form, so they are not considered unique. The answer should be a function of n.

Cubes as Differences of Squares [07/04/2002]
Prove that the cube of any positive integer is equal to the difference of the squares of two integers.

Cubic Diophantine Equation in Three Variables [09/10/2004]
Find integer solutions of the equation x^3 + y^3 = 31z^3. I know the fundamental solution is (137, -65, 42), but I want to have all the values positive. I know also that there is an arithmetic procedure (doubling in the group) to obtain further solutions from the fundamental one, but I do not know the details of this procedure.

Cyclic Redundancy Check [06/26/2002]
I understand how cyclic redundancy checks work, but I fail to see how appending zeros to the message string (before the division) provides an advantage.

Decomposition of Primes as the Sum of Two Squares [12/17/2005]
If you write a prime number p as a sum of two squares, can you prove that its decomposition a^2 + b^2 is unique for (a,b up to +/- 1)?

Degree of Error in pi(x) Approximation [10/20/2004]
How does the error in the formula x/ln(x), used to approximate pi(x) (the primes counting function), behave for large values of x?

Determine b if (a+b)/5 = (b-1)/2 [07/25/2003]
Both a and b are 4 four-digit numbers, a is greater than b, and one number is obtained from the other by reversing the digits. Determine b if (a+b)/5 = (b-1)/2.

Difference of Square Numbers [07/18/2008]
Can one number ever be represented as two distinct differences of squares? Or is every difference of square numbers unique?

A Diophantine Determined with a Data-Driven Approach [06/24/2012]
A student wonders if a u-substitution would solve a troublesome Diophantine equation. Doctor Vogler takes up the struggle with a false start of his own, before noticing helpful patterns from a computational attack and walking the student through the thought process behind his strategy.

Diophantine Equation [08/25/2007]
Find all integer solutions (a,b) such that (1-ab-a-b)/(1-ab+a+b) is an integer.

Diophantine Equations in Three Variables [10/30/2004]
I need to know how to get positive integer solutions of two Diophantine equations having three variables. For example: 2x + 3y + 7z = 32 ; 3x + 4y - z = 19. (Give the positive set of triples for the above equations.)

Diophantine Equation Solved by Reasoning [04/05/2006]
Find all integer solutions of (1/m) + (1/n) - (1/mn^2) = 3/4.

Diophantine Equations, Step by Step [10/01/2002]
Find all positive integer solutions to 43x + 7y + 17z = 400.

Diophantine Equation with Three Unknowns [09/28/2009]
Find all triples of positive integers (x,y,z) such that xyz + xy + yz + zx + x + y + z = 243.

Diophantine Equation with Three Variables [02/01/2006]
Find all positive integer solutions (a,b,c) that satisfy the equation a + b + c + ab + bc + ca = abc + 1.

Diophantine System of Equations [12/31/2004]
I'm trying to find the rational solutions (in parametric form) to the set of two simultaneous equations: x^2 + x = y^2 x^2 + 1 = z^2

Direct Conversion of Binary to Octal [05/14/2002]
How can you convert from base 2 to base 8 without going through base 10?

Discrete Logarithm Problem [10/13/2004]
Given a === b^c mod N. When a, b, and N are given, can we find c?

Distances between Rational and Irrational Numbers [01/13/2005]
Is it true that if |x - p/q| < |x - r/s| with x irrational and p/q, r/s irreducible rationals then q > s? If so, how can you prove it?

Divisibility by 8 [04/14/1997]
Show that, if n is a positive integer, then 5^n+2*3^(n-1) + 1 is divisible by 8.

Divisibility Proof [02/16/2001]
How can I prove that if n is an odd positive integer, then 2269^n + 1779^n + 1730^n - 1776^n is an integer multiple of 2001?

Divisibility Proof by Cases [02/23/2003]
Prove that if d|n, then (2^d - 1)|(2^n - 1).

Do Rational and Irrational Numbers Alternate? [10/13/2000]
If any two non-equal real numbers "contain" an irrational, and any two non-equal real numbers "contain" a rational, do rational and irrational numbers alternate?

Elliptic Curve Factorization [02/10/2006]
I would like to find out how to develop the parameters of a cubic parabola in general so that I can implement an integer factorization method. I would also like to know how to add points such as P+P and P+Q to such a curve.

Elliptic Curve Resources [05/04/2007]
Is there an algorithm to determine the rank of a general elliptic curve? What are some good books to start learning about elliptic curves?

Equation with Two Exponential Terms [06/27/2009]
Find all ordered pairs (a,b) for which 3^a + 7^b is a perfect square.

Erdos' Proof [04/03/1997]
Can you show me Erdos' proof that there is a prime number between every integer n and 2n?

Euclidean Algorithm [5/13/1996]
How can we prove that Euclid's method for finding the highest common factor for two numbers will work for all values?

Euclidean Algorithm and Linear Equations [11/03/2003]
Could you please explain step by step how to use the Euclidean Algorithm to solve a linear equation and find x and y integers?

Euclidean Algorithms [3/13/1996]
What is the Euclidean algorithm? What is a "constructible" number? What can you tell me about Diophantine equations?

Euclidean and Division Algorithms [11/26/1997]
Can you show and explain the proofs of the Euclidean Algorithm and the Division Algorithm?

Euclid's Extended Algorithm [09/16/2001]
Can you please state for me the steps of Euclid's extended algorithm in simple terms?

Euler's Phi Function Applied to Large Numbers [08/23/2004]
I'm trying to find the phi value of a large (10-digit) composite number. Can it be done in polynomial time, and if so, is there an algorithm?

Expansion of n(n+1)(n+2)...(n+k) [07/15/2002]
Is the expansion of n(n+1)(n+2)...(n+k) known?

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