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

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.

Stirling's Formula [05/17/2003]
I am having trouble finding an algorithm to solve the following problem: What's the least number for x such that x! >= 3^x is true?

Summing Activity Leads to a Mean of e [04/01/2005]
I asked my students to keep adding random integers from 1 to 100 until the sum exceeded 100. We then found the average number of terms added. The answer seems to be e. Why? The more we do it, the closer we get.

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