About XGC

This web site presents XGC(c,m).

GCthe Goldbach Conjecture
XGC(c,m)the eXtended Goldbach Conjecture of residue c and modulus m
ESthe Eratosthenes Sieve
XESthe eXtended Eratosthenes Sieve

XGC(c,m) is a generalization of GC by means of two parameters: c and m.
This way GC becomes the instance of XGC(c,m) for c=1 and m=2, or GC = XGC(1,2)

As well as GC is about numbers of the Primes set, so XGC(c,m) is about numbers of the Euclid(c,m) set.
Note that the Odd Primes set is the instance of the Euclid(c,m) set for c=1 and m=2, or OddPrimes = Euclid(1,2) and it's so by construction, not by definition. In fact, as well as prime numbers are found by ES applied to natural numbers, so Euclid(c,m) numbers are found by XES applied to the numbers congruent to the residue c of the modulus m.

The following table shows GC and XGC side by side, so that you can easily compare them. As you see, XGC is a simple generalization of GC. You should try it by yourself.

GC: every natural n = 2 k, where k >= 1 + 2 , is the sum of 2 numbers of the Euclid(1, 2) set
XGC: m c + m m Euclid(c, m)
keywords
Mathematics, Number Theory, MSC 11P32, Unsolved Problems, Goldbach Conjecture, Eratosthenes Sieve, Mathematica, Sequence Algebra, Goldbach-type theorems

Formulas and formulations are given by means of Mathematica functions.

 

History of XGC

Enjoy yourself!

Andrea