See also the
Dr. Math FAQ:
0.9999 = 1
0 to 0 power
n to 0 power
0! = 1
dividing by 0
Browse College Number Theory
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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
- 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
- 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
- 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,
- 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
- 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
- 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.