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Number theory

This site features:

1. Generic Two integer variable equation solver: Diophantine equation ax² + bxy + cy² + dx + ey + f = 0 solver, where the unknowns x and y can be integer numbers only. Written in Java/JavaScript. Last updated on May 31st, 2001.

2. Quadratic modular equation solver: Calculator that can solve equations of the form ax² + bx + c = 0 (mod n). Last updated: May 2nd, 2002.

3. Sum of powers: Table of relations of the form a^p + b^q = c^r with gcd(a,b,c) = 1. Last updated: September 7th, 2003.

4. Ulam's Spiral: Java applet featuring a graphical view of prime numbers. Last updated on February 14th, 2003.

5. Factorization using the Elliptic Curve Method: Applet that can be used to find 20- or 30-digit factors of numbers or numerical expressions up to 1000 digits long. It also computes the number and sum of divisors, Euler's totient and Moebius, and its decomposition as a sum of up to 4 perfect squares. Last updated on February 9th, 2003.

6. Gaussian Integer Factorization applet: Finds the factors of complex numbers of the form a+bi where a and b are integers. It also includes a complete calculator with operators and functions using gaussian integers. Last updated on June 1st, 2002.

7. Gaussian Primes: Java applet featuring a graphical view of gaussian prime numbers. Last updated on June 15th, 2003.

8. Discrete logarithm calculator: Applet that finds the exponent in the expression Base^Exponent = Power (mod Modulus). Last updated on March 30th, 2002.

9. Continued fraction calculator: This calculator can find the continued fraction expansions of rational numbers and quadratic irrationalities. Last updated on April 28th, 2002.

10. Every positive integer is a sum of four integer squares: Constructive proof of this interesting theorem. Last updated: October 5th, 2001.

11. Sum of squares: This calculator can find the decomposition of a number or numerical expression in a sum of up to four squares. It does not need its prime factorization. Last updated on November 27th, 2002.

12. Sum of four cubes: This calculator can find the decomposition of a number or numerical expression that is not congruent to 4 or 5 (mod 9) in a sum of four cubes. Last updated on March 27th, 2005.

13. Brilliant numbers: Interesting problem about products of two primes of the same size.

14. Factors of Modified Fermat Numbers: Factors of numbers of the form 4³ⁿ+2³ⁿ+1 and 4³ⁿ-2³ⁿ+1.
    
    

15. Factors of numbers near googolplex (10^{10^100}):

    • Table 1: Factors of numbers in the range 10^{10^100} to 10^{10^100} + 999.

    • Table 2: Factors of numbers in the range 10^{10^100} - 999 to 10^{10^100}.

    • Table 3: Factors of numbers of the form 10^{10^100} + 10^Exp, where 0 <= Exp <= 9.

16. Factors of numbers near googolplexplex (10^googolplex):

    • Table 1: Factors of numbers in the range 10^{10^(10^100)} to 10^{10^(10^100)} + 999.

    • Table 2: Factors of numbers in the range 10^{10^(10^100)} - 999 to 10^{10^(10^100)}.

Cool math site of the week
August 30th, 2002.