Section 1.1: The Nature of Conjectures

A conjecture is essentially a statement that lacks a formal proof. Typically, our intuition suggests that the conjecture is true, yet we are unable to substantiate it.

1. Twin Prime Conjecture (Euclid, circa 300 BC)

Twin primes are defined as pairs of primes ( p ) and ( p + 2 ). For instance, pairs like 11 and 13 or 17 and 19 serve as examples of twin primes. The conjecture posits:

• Conjecture: There are infinitely many twin primes of the form ( p ) and ( p + 2 ).

Despite widespread belief among mathematicians about its validity, no proof has been established. As numbers increase in size, primes become less frequent, making twin primes exceptionally rare. This conjecture is frequently referred to as Euclid’s Twin Prime Conjecture. While Euclid provided the earliest proof of the infinitude of primes, he did not address twin primes directly.

Proof that there are infinitely many primes! Euclid’s proof is considered to be one of the most beautiful.

In a groundbreaking development in 2013, mathematician Yitang Zhang demonstrated that there exists an infinite number of prime pairs ( p ) and ( q ) with a difference of less than 70 million:

• ( q - p < 70,000,000 )

This significant finding marked the first time a finite limit on the distance between primes was established. Subsequent improvements by various mathematicians have tightened this bound to just 246, meaning:

• ( q - p < 246 )

Interestingly, there also exist pairs of primes ( p ) and ( q ) that are only 2 units apart, but whether such pairs are finite or infinite remains unresolved.

2. Lagrange’s Conjecture (1775)

• Conjecture: Every odd integer greater than 5 can be expressed as ( p + 2q ), where ( p ) and ( q ) are both primes.

For example:

• ( 7 = 3 + 2(2) )
• ( 9 = 5 + 2(2) )
• ( 11 = 5 + 2(3) )

Once again, no proof exists for this conjecture.

3. Goldbach’s Conjecture (1642)

Christian Goldbach, who lived from 1690 to 1764, proposed another notable unproven conjecture in number theory:

• Conjecture: Every even number greater than 2 can be expressed as the sum of two primes.

For instance:

• ( 4 = 2 + 2 )
• ( 6 = 3 + 3 )
• ( 8 = 5 + 3 )

Although this conjecture has been verified for all even integers up to ( 4 times 10^{18} ) using modern computational methods, a general proof remains elusive. The closest proven statement is that every even integer greater than 2 can be expressed as the sum of six or fewer primes. There is also an alternative Goldbach Conjecture, known as the Odd Conjecture, which states:

• Every odd integer greater than 5 can be expressed as the sum of three primes.

Until recently, this conjecture was only validated for odd numbers exceeding ( 2 times 10^{1346} ) — a staggering number comprising 1,347 digits! However, in 2013, mathematician Harold Helfgott confirmed its truth for all odd integers greater than 5. Both Goldbach conjectures highlight the challenges of number theory; they are straightforward to articulate but complex to resolve.

4. Landau’s Problems (1912)

At the International Congress of Mathematicians in 1912, Landau outlined four fundamental problems concerning prime numbers. Three of these are the aforementioned conjectures. The fourth problem asserts that:

• There are infinitely many primes of the form ( n^2 + 1 ).

For example, ( 2^2 + 1 = 5 ) and ( 4^2 + 1 = 17 ). As of September 2021, all four problems remain unsolved. Remarkably, mathematicians have made significant strides toward addressing these conjectures. It is hoped that someday definitive proofs will emerge.

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