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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 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.

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 numbers?

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 prime?

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 efficient method?

Transfinite Arithmetic [10/28/1997]

What is transfinite arithmetic? I pretty much know what it means, but I am having trouble applying it to aleph-null.

Transfinite Numbers [11/07/1997]

I know that Georg Cantor discovered transfinite numbers, but what are they?

Triangle Perimeters [12/15/1998]

How many triangles with integer sides have a given perimeter? How does the triangle inequality enter into the proof?

Triangular Numbers [04/03/1997]

How do I prove that there are an infinite number of triangular numbers that are equal to a square number?

Tribonacci Numbers [11/11/2000]

Is there an implicit formula to calculate the nth Tribonacci number? Also, is there a formula to find the sum of the first n Tribonacci numbers?

Unique Decomposition of Pythagorean Primes [05/19/2002]

Is it true that a Pythagorean prime (i.e., a natural prime that can be expressed as a sum of squares of two integers) can be expressed as a sum of two squares in one and only one way?

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