See also the
Dr. Math FAQ:
0.9999 = 1
0 to 0 power
n to 0 power
0! = 1
dividing by 0
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Selected answers to common questions:
Testing for primality.
- 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 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.
- 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.
- Sum of 1/Sqrt(i) [11/20/2000]
What is the formula for the sum of 1/sqrt(i) for i = 1 to n? Can you show
me the proof by induction?
- Sum of Distinct Fibonacci Numbers [05/06/2001]
How do you show that every positive integer is a sum of distinct terms of
the Fibonacci sequence?
- Sum of First N Positive Integers Making a Perfect Square [02/28/2006]
For what integer values of n and k does 1 + 2 + 3 + ... + n = k^2?
- Sum of Quadratic Residues [07/22/1999]
Given a prime number p = 4k+3. If d = (sum quadratic nonresidues) - (sum
quadratic residues) in (0,p) can we prove that d is greater than 0 for
all such p? Can we prove that d goes to infinity when p goes to infinity?
- Sum of Sequence: a, b, a+b, a+2b, 2a+3b... [01/26/2003]
Find the sum of the first 30 terms of the sequence: 1, 5, 6, 11, 17,
28... if the 30th term is 2888956 and the 31st term is 4674429.
- Sum of Two Squares [05/26/2003]
Can you generate the sequence [400, 399, 393, 392, 384, 375, 360, 356,
337, 329, 311, 300]?
- Sums of Consecutive Numbers [06/20/2002]
In what way(s) can 1000 be expressed as the sum of consecutive
- System-Level Programming and Base 2 [05/03/2001]
In computer programming, I have a result that contains several values,
always a power of 2 (2^2, 2^3, 2^4). If my value is 2^3, 2^4, 2^6 304,
how can I tell if 2^3 exists in 304?
- Testing primality of 32-bit numbers [11/18/1994]
What is the best (fastest) way to test if an arbitrary 32-bit number is
- Testing Prime Numbers [05/12/2003]
Besides the Sieve of Eratosthenes, what other methods can be used to
determine all prime numbers within a given range? Is there a more