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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.



Relationship Between GCF and LCM [05/22/2002]
What is the exact relationship between the gcf or gcd and the lcm of two numbers?

Relatively Prime Pythagorean Triples [09/13/1997]
Questions about Pythagorean triples.

Remainders, Pigeons, and Pigeonholes [03/26/2003]
Given 17 integers, prove that it is always possible to select 5 of the 17 whose sum is divisible by 5.

Remainder when Dividing Large Numbers [04/17/2001]
How can I find the remainder when (12371^56 + 34)^28 is divided by 111?

Repeating Decimals [05/14/1997]
If the length of the repeating sequence in a decimal of a converted fraction is less than the denominator of the fraction, is it always an integer factor of the denominator minus one?

Repeating Digits of Fractions [04/28/1999]
Do you know any theorems relating to the length of the repeating portion of the decimal representation of fractions?

Representing Positive Integers in an Irrational Base [08/13/2007]
I know that 3 in base 2 is written as 11. But how would I express 3 in terms of an irrational base, like base square root of 2?

Residues and Non-Residues [05/04/2003]
If p > 3, show that p divides the sum of its quadratic residue.

Reverse Modulus Operator [10/09/2001]
Is there an operator that would return 2 when we we do 6 * 0, * being this new operator?

Roots of Unity Exactly [02/13/2011]
A student seeks expressions for roots of unity in terms of rational operators and real-valued roots. After revealing a constraint associated with Fermat primes, Doctor Vogler writes out the first few roots of unity, outlines the general idea -- invoking the half-angle formulas for sine and cosine and reducing the problem to a series of quadratic equations to solve -- then demonstrates the method for a seventeenth root of unity.

RSA Encryption [04/25/2002]
Decrypt the encrypted message in ciphertext C to find the original plaintext, a string of English letters.

Second-Order Linear Recurrences [06/08/2001]
Three problems involving recurrence equations.

Second Order Recurrence with Non-Constant Coefficients [05/27/2005]
I'm trying to find a closed form solution of a second order recurrence relation with no constant coefficients, specifically: u(n+2) = 2*(2*n+3)^2 * u(n+1) - 4*(n+1)^2*(2*n+1)*(2*n+3)*u(n). Can you help?

Shanks-Tonelli Algorithm [01/17/2001]
How can I calculate the four square roots of a number modulo n, where n is the product of two primes p and q?

Showing Two Numbers Are Relatively Prime [08/01/2008]
Show that for every natural number n, 21n + 4 and 14n + 3 are relatively prime.

Simultaneous Modulus Congruencies [04/18/2001]
How can I find x if x = 3 (mod 8), x = 11 (mod 20) and x = 1 (mod 15)?

Slot-wise Addition of Pythagorean Triples [07/17/2003]
Is it possible to have a primitive Pythagorean triple, (a,b,c) such that a^2+b^2 = c^2, and two other Pythagorean triples, not necessarily primitive (x,y,z) and (p,q,r) with the property that a=x+p, b=y+q, and c=z+r?

Solutions for Pell's Equation [12/11/2000]
Do you know of an algorithm to find solutions for Pell's equation (x^2 - Dy^2 = 1) for different values of D?

Solving a Diophantine Equation [05/01/2005]
How can I find all integer solutions of an equation in the form aXY + bX + cY + d = 0? For example, 5XY + 3X - 8Y - 8 = 0.

Solving a Diophantine Equation by Use of Elliptic Curves [03/23/2008]
How can I solve the Diophantine equation 4u^3 - v^2 = 3 or others similar to it?

Solving a Diophantine Equation By Use of Number Fields [03/01/2006]
Prove that the equation x^2 = y^7 + 7 has no integer solutions (x,y).

Solving a^n + b^n = c^n [03/10/2009]
Fermat's Last Theorem says there are no positive integer solutions to a^n + b^n = c^n for n > 2. How do I find solutions for n <= 2?

Solving an Exponential Diophantine Equation with Modular Arithmetic [11/23/2005]
Find all positive integer solutions a, b, c to (5^a)*(7^b) + 4 = 3^c.

Solving a Nonlinear Diophantine Equation [10/30/2005]
Given x^3 = 3y^2 + 3y + 1 where x < y, are there integer solutions for x and y?

Solving a Quartic Diophantine Equation [08/20/2005]
I would like to know if there is a method of solving the equation x^4 + 14*x^2 + 1 = y^2 in rationals.

Solving a Quartic Diophantine Equation [04/23/2008]
Find all integer solutions other than 0 to the Diophantine equation x^4 + 2191x^3 + 1931x^2 + 1037x + 6754801 = y^2.

Solving ax^2 + by + c = 0 Using Modular Arithmetic [10/09/2004]
How can I find integer solutions for x and y with an equation of the form ax^2 + by + c = 0?

Solving Cubic and Quartic Polynomials [04/30/1998]
Could you describe the algorithms used to solve cubic and quartic polynomials (Tartaglia's Solution)?

Solving Diophantine Equations [12/20/2004]
General strategy for solving Diophantine equations and a specific strategy for the equation xyzw + 3xyz + 3xy + 3x = c.

Solving Diophantine Equations [04/18/2008]
I have recently delved into the world of Diophantine equations. I've found some solutions to the equation 7x^2 + 1 = y^3. Can you show me how to find others or to determine that I have found them all?

Solving More Exponential Diophantine Equations with More Modular Arithmetic [02/02/2011]
Doctor Vogler applies modular arithmetic, as well as the Wieferich prime, to approach several exponential Diophantine equations and expressions.

Solving Multivariable Diophantine Equations [05/03/1998]
Finding general solutions to two diophantine equations.

Solving Quadratic Diophantine and Pell Equations [05/04/2005]
Would you show me how to solve Diophantine equations that have the form a*x^2 + b*y^2 = z^2, with a, b, and z given?

Solving System of Equations Using Elliptic Curves [12/15/2006]
Find a rational number s such that s^2 + 7 and s^2 - 7 are both squares of rational numbers.

Solving the Diophantine Equation x^y - y^x = x + y [04/30/2005]
Find all integer solutions of x^y - y^x = x + y.

Solving with the Pell Equation [11/15/2004]
In class we are discussing how to solve problems of the form x^2 - ny^2 = N for fixed (n,N). The process is fairly simple if |N| < n^(1/2); however, how does one find solutions if |N| > n^(1/2)? Specifically, how does one extract integer solutions to x^2 - 29*y^2 = 7 with x and y being integers greater than 1000?

Spacing between Prime Numbers [11/08/2005]
Where is the first place that the difference between two consecutive prime numbers exceeds 2000? Is there a formula or general approach to finding such differences without having to just read through lists of known primes?

The Square Root of i [05/25/1997]
What is the square root of i?

Stirling Numbers [05/26/1999]
Can you show how to evaluate Stirling Numbers of the first and second kinds?

Stirling Numbers of the Second Kind, Bernoulli Numbers [05/29/2001]
Sk = 1^k+2^K+3^k+...+n^k. Find Sk as a formula.

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