twin primes paradox

[AntonioLao]

Analogous to the twin paradox of special relativity, the twin primes paradox is one founded in the mathematics of pure numbers. Specifically, it is found in the prime distributions for the set of whole numbers. The former is a physical paradox while the latter is an abstract paradox.

The first occurrence of twin primes is the pair (5,7), the second is (11,13), the third is (17,19), the 4th is (29,31), the 5th is (41,43), the 6th is (59,61), the 7th is (71,73), the 8th is (101,103), the 9th is (107,109), the 10th is (137,139), the 11th is (149,151), the 12th is (179,181), the 13th is (191,193), the 14th is (197,199), the 15th is (227,229), the 16th is (239,241), the 17th is (269,271), the 18th is (281,283), the 19th is (311,313), the 20th is (347,349), the 21st is (419,421), the 22nd is (431,433), the 23rd is (461,463), the 24th is (521,523), the 25th is (569,571), the 26th is (599,601), the 27th is (617,619), the 28th is (641,643), the 29th is (659,661), the 30th is (809,811), the 31st is (821,823), the 32nd is (827,829), and many more. It can be noted that the difference is always 2 and their occurrence seems endless.

Twin primes could belong to one of Erdös conjectures regarding prime numbers. This particular conjecture states that every even number that is the product of the one and only even prime number 2 and another comparably smaller prime is also equal to the difference of two bigger primes. In the form of a simple equation, Erdös prime difference conjecture is given by a=½(c-b) where a, b, and c are all prime numbers such that a<b<c. The conjecture is that there are infinite pairs of (b,c) such that the simple equation is always true. For the case of twin primes, the even number is 2 and the smallest prime must be positive unity. Unfortunately, positive unity is not considered as a prime number. The next even number is 4 which is the product of 2 and itself. The other two bigger primes also seem to occur endlessly, for examples: 4=7-3, 4=11-7, 4=17-13, 4=23-19, 4=41-37, 4=71-67, 4=83-79, 4=101-97, 4= 107-103, 4=113-109, so on and so forth. From these, the twin primes paradox becomes the question whether positive unity or the whole number 1 is a prime? It must in order to raise Erdös conjecture a step closer to becoming a theorem. On the other hand, since positive unity does not appear in the sieve of Diophantus while the sieve does give probable locations of all pairs of twin primes, the most logical conclusion is to exclude positive unity as a prime number and retain the sieve for triple algorithmic proofs of Erdös, Euler, and Goldbach conjectures on prime numbers.

[SteveA]

I've seen some other things that similarly make it questionable whether or not 1 should be considered a prime number.

Maybe we need a different set of primes? (For example, by changing multiplication)

Notice that multiplication takes two, 1 dimensional values and constructs an area from them, but we don't need to use squares ... you only need 3 points to define a triangle and we can tile surfaces with triangles that may not be tilable with squares.

There was a pseudo method of multiplication that constructed surfaces as the difference between two triangular areas (the end result is a trapezoid).

It had an interesting property that it could not construct powers of 2 unless it already possessed power of 2 lengths, so the set of powers of 2 was similar to a set of primes because you could not construct a prime via "multiplication" without already having a prime number.

A drawback was that it wasn't commutative, but if you're interested at all, the formula used was q=(a^2+a)/2-(b^2-b)/2 and it's similar to computing multiplication as a difference between two squares ab=((a+b)^2-(a-b)^2)/4.