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.
- 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.
- 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?