K = A² – B³ (MORDELL’S EQUATION)

K  =  A² – B³  (MORDELL’S EQUATION)

The program finds the solutions of the equation  K  =  A² – B³  with  K,A,B integers.

DOWNLOAD:

http://numbertheorycalculator.myblog.it/k-a-b%C2%B3/

AMICABLE NUMBERS

AMICABLE NUMBERS.

Amicable numbers are two different numbers so related that the sum of the proper divisors of one of the numbers is equal to the other. (A proper divisor of a number is a positive integer divisor other than the number itself).

DOWNLOAD:

http://numbertheorycalculator.myblog.it/amicable-numbers/

ALIQUOT SEQUENCES

An aliquot sequence is a recursive sequence in which each term is the sum of the proper divisors of the previous term.
There are 5 cases:
1) The sequence ends with 1.
2) The sequence ends with the starting number and then repeats (Sociable Numbers).
3) The sequence is made by only two numbers (Amicable Numbers).
4) The sequence is made by only one number (Perfect Number).
5) Some numbers have an aliquot sequence which is eventually periodic, but the number itself is not perfect, amicable, or sociable (ex. 95).

DOWNLOAD:

http://numbertheorycalculator.myblog.it/aliquot-sequences/

LEAST QUADRATIC NON RESIDUE AND LEAST PRIMITIVE ROOT

LEAST QUADRATIC NON RESIDUE AND LEAST PRIMITIVE ROOT OF A PRIME NUMBER.

DOWNLOAD:

http://numbertheorycalculator.myblog.it/least-qnr-and-least-pr/

QUADRATIC CONGRUENCE

QUADRATIC CONGRUENCE.

The program checks if A is quadratic residue or quadratic non-residue of P. If A is quadratic residue of P, the program solves the congruence:
X² ≡ A  (modulo P)
Namely the program finds the two X values less than P, such that A is the remainder of dividing  X² for P.

DOWNLOAD:

http://numbertheorycalculator.myblog.it/quadratic-congruence/

ORDER OF A MODULO P

ORDER OF A MODULO P

The order of A modulo P (prime number) is the smallest positive R for wich : A^R ≡ 1 (mod. P). Since A^(P-1) ≡ 1 (mod. P) always, it is obvious that, if the order of A is less than (P-1), the order should divide (P-1). If the order of A modulo P is equal to (P-1), then A is a primitive root of P.

[ A^R ≡ 1 (mod. P)  means that the rest of the division of A^R divided by P is 1 ]

DOWNLOAD:

http://numbertheorycalculator.myblog.it/order-of-a-modulo-p/

SOPHIE GERMAIN NUMBERS

SOPHIE GERMAIN NUMBERS IN A GIVEN RANGE.

A PRIME NUMBER P IS A SOPHIE GERMAIN PRIME IF 2P + 1 IS ALSO PRIME.

DOWNLOAD:

http://numbertheorycalculator.myblog.it/sophie-germain-numbers/

CONTINUED FRACTION OF A SQUARE ROOT

CONTINUED FRACTION OF A SQUARE ROOT.

THE SQUARE ROOT OF A SQUAREFREE INTEGER HAS A PERIODIC CONTINUED FRACTION OF THE FORM:

a0, a1, a2, a3, a4………..a2, a1, 2a0

DOWNLOAD:

http://numbertheorycalculator.myblog.it/continued-fractions/

PRIME DESERTS

PRIME DESERTS.

THE PROGRAM FINDS THE LARGEST INTERVAL OF CONSECUTIVE NON-PRIMES IN A GIVEN RANGE  A – B.

DOWNLOAD:

http://numbertheorycalculator.myblog.it/prime-deserts/

FRACTIONS GENERATING SQRT(2)

FRATIONS GENERATING SQRT(2).

START WITH  1/1 , THEN FROM EACH FRACTION  A/B , THE FOLLOWING WILL BE  (A + 2B)/(A + B).

DOWNLOAD:

http://numbertheorycalculator.myblog.it/sqrt-2/

PI DIGITS

PI DIGITS.

THE PROGRAM CALCULATES THE DIGITS OF PI.

DOWNLOAD:

http://numbertheorycalculator.myblog.it/pi-digits/

DIOPHANTINE EQUATION

DIOPHANTINE EQUATION:

Ax + By = C

A,B,C,x,y  POSITIVE INTEGERS.

DOWNLOAD:

http://numbertheorycalculator.myblog.it/diophantine-equation/

QUADRATIC RESIDUES

QUADRATIC RESIDUES OF A PRIME NUMBER.

A quadratic residue of a prime number is the remainder of the division of a perfect square for the prime number. All prime numbers have (p-1) / 2 quadratic residues and (p-1) / 2 quadratic non residues.

DOWNLOAD:

http://numbertheorycalculator.myblog.it/quadratic-residues/

X^2 – x + 41

STUDY OF THE POLYNOMIAL  X^2 – X + 41

THE POLYNOMIAL  X^2 – X + 41 GENERATES 40 CONSECUTIVE PRIMES FROM X = 1 TO X = 40. THE PROGRAM CALCULATES THE NUMBER OF PRIMES GENERATED BY THIS POLYNOMIAL IN A GIVEN RANGE.

DOWNLOAD:

http://numbertheorycalculator.myblog.it/x-2-x-41/

PASCAL TRIANGLE

PASCAL TRIANGLE.

THE PROGRAM DISPLAYS THE N-TH ROW OF PASCAL’S TRIANGLE.

DOWNLOAD:

http://numbertheorycalculator.myblog.it/pascal-triangle/

PRIMALITY TEST

PRIMALITY TEST.

THE PROGRAM CHECKS WHETHER AN INTEGER IS A PRIME NUMBER OR NOT.

DOWNLOAD:

http://numbertheorycalculator.myblog.it/archive/2010/04/03/primality-test.html

NEXTPRIME

NEXTPRIME.

THE PROGRAM FIND THE SMALLEST PRIME NUMBER GREATER THAN A GIVEN NUMBER.

DOWNLOAD:

http://numbertheorycalculator.myblog.it/archive/2010/04/02/nextprime.html

DIVISORS

THE PROGRAM CALCULATES THE PROPER DIVISORS OF AN INTEGER, THEIR SUM AND THEIR NUMBER.

DOWNLOAD:

http://numbertheorycalculator.myblog.it/divisors/

PRIME NUMBERS

PRIME NUMBERS IN A RANGE.

DOWNLOAD:

http://numbertheorycalculator.myblog.it/prime-numbers/

PRIME FACTORS

PRIME  FACTORS  OF  AN  INTEGER.

DOWNLOAD:

http://numbertheorycalculator.myblog.it/prime-factors/

PYTHAGOREAN TRIPLES

Pythagorean triple: consists of three positive integers A, B, and C, such that A²  =  B²  + C².
Euclid’s formula is a fundamental formula for generating Pythagorean triples given an arbitrary pair of positive integers M and N with M > N. The formula states that the integers:

A  =  M²  +  N²
B  =   M²  -  N²
C  = 2*M*N

form a Pythagorean triple.

DOWNLOAD:

http://numbertheorycalculator.myblog.it/pythagorean-triples/

GOLDBACH CONJECTURE

GOLDBACH CONJECTURE: is one of the oldest unsolved problems in Number Theory. It states: every even number greater than 2 can be expressed as the sum of two primes.

DOWNLOAD:

http://numbertheorycalculator.myblog.it/goldbach/

TWIN PRIME PAIRS

TWIN PRIME PAIRS.

DOWNLOAD:

http://numbertheorycalculator.myblog.it/twin-prime-pairs/

COLLATZ CONJECTURE

COLLATZ CONJECTURE:

Take any natural number n. If n is even, divide it by 2 to get n / 2, if n is odd multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1.

DOWNLOAD:

http://numbertheorycalculator.myblog.it/collatz-conjecture/

N = X^2 + Y^2

THE PROGRAM SOLVES  N = X^2 + Y^2 (N,X,Y INTEGERS).

DOWNLOAD:

http://numbertheorycalculator.myblog.it/n-x-2-y-2/

MODPOW: A^B MODULO C

MODPOW. THE PROGRAM CALCULATES  A^B MODULO C.

DOWNLOAD:

http://numbertheorycalculator.myblog.it/modpow/

FIBONACCI NUMBERS

THE PROGRAM CALCULATES FIBONACCI NUMBERS:

DOWNLOAD:

http://numbertheorycalculator.myblog.it/archive/2010/03/28/fibonacci-numbers.html