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Racing Prime Conjectures With Nuances...And a Recent Discovery

One can always find news about the prime numbers, as they seem to inherently contain richness yet undiscovered. Yet, we must be careful in making conjectures, as we really do not understand primes very well....especially since we also know that there are an infinite number of them. But, the race for knowledge is on!

For example, two famous conjectures involving primes are:

  • Goldbach's Conjecture (1742): Every even integer greater than 2 can be expressed as the sum of two primes. For example, 10 = 3 + 7 or 12 = 5 + 7.
  • Twin-Prime Conjecture: There are infinitely many primes p such that p + 2 is also prime. For example, 11 and 13 or 17 and 19.
Now, considerable work has been done on both Conjectures. As to the first, it is true for even integers less than 4 × 1018...and probably is true for all (though no proof exists and exhaustion techniques are not an option).

The second conjecture also is elusive. As of 2011, the largest known prime pair was 3756801695685 * 2666669 ± 1, numbers which have 200,700 digits.

A common technique is to use large numbers such as n = 1,000,000,001! to produce the string of 1 billion composite numbers via the sequence n+2, n+3, n+4, ..., n+1,000,000,001. That is, this string includes no twin primes.

In discussing this lengthy string of composite numbers, Frank Hudson (see below) notes: "Of course, this same approach could be used to construct a sequence of consecutive composite numbers having any desired length...Thus, the length of the gaps between primes tends to infinity."

A MAA review of this article notes that this claim is "undemonstrated and incorrect"! The review continues: "Indeed, if one did prove that the gap between successive primes does tend to infinity, then a corrollary would be that the twin prime conjecture is false."

To muddy the waters, one needs to consider the related Hardy–Littlewood Conjecture, . As a generalization of the Twin Prime Conjecture, it focuses on the distribution of prime "constellations" (which includes twin primes) in a manner related to the famous Prime Number Theorem.

This week, Yitang Zhang, an unheralded mathematician at the University of New Hampshire, made a significant step forward in our knowledge of twin primes. According to Peter Sarnak, a lauded mathematician at Princeton, Zhang's discovery is "a deep insight. It's a deep result."

Zhang proves the existence of an infinite number of prime pairs whose difference is less than a fixed value, currently set at 70 million. Now, given Zhang's work, the thrust is to narrow this difference down to 2!

I like these fine distinctions and fascinating connections...that we can construct arbitrarily long sequences of composite numbers or that there exist an infinite number of twin primes given a larger difference than 2. Neither fact implies that the gap between successive primes tends to infinity...nor that there are an infinite number of twin primes....yet! It is like a race...which mathematical idea will win. Isn't mathematics wonderful!

Source: F. Hudson's "Are the Primes Really Infinite?" Mathematics Teacher, November 1990.