Two conjectures about primes

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Subsection 5.3.2 Two conjectures about primes

Though we were able to prove there are infinitely many primes, there are several open questions about primes.

Definition 5.3.2.1.

Twin primes are pairs of primes that differ by \(2\text{.}\)

Example 5.3.2.2. Some twin primes.

These are twin prime pairs:

\(3\) and \(5\text{,}\) \(5\) and \(7\text{,}\) \(11\) and \(13\text{,}\) \(17\) and \(19\)

It is conjectured that there are infinitely many twin primes.

Conjecture 5.3.2.3. Twin prime conjecture.

There are infinitely many primes \(p\) such that \(p + 2\) is also prime.

While the twin prime conjecture is still open, there have been several amazing recent advances in this direction. See the video Gaps between Primes

¹

www.youtube.com/watch?v=vkMXdShDdtY

posted by Numberphile for some of this story.

Another famous open problem in number theory concerns decomposing integers into sums of primes. In a letter to Leonhard Euler in 1742, Christian Goldbach conjectured that every odd integer \(n\text{,}\) \(n \gt 5\text{,}\) is the sum of three primes. Euler replied that this conjecture is equivalent to the conjecture that every even integer \(n\text{,}\) \(n>2\text{,}\) is the sum of two primes. For example, \(4 = 2 + 2\text{,}\) \(6 = 3 + 3\text{,}\) \(8 = 3 + 5\text{,}\) \(10 = 3 + 7\text{,}\) and so on.

Although no proof of the Goldbach conjecture has been found, the conjecture has been computationally checked to hold up to \(4 \cdot 10^{18}\text{,}\) see [4].

Conjecture 5.3.2.4. Goldbach conjecture.

Every even integer \(n\text{,}\) \(n \gt 2\text{,}\) is the sum of two primes.

Figure 5.3.2.5. Goldbach Conjectures ({https://xkcd.com/1310/}) The weak twin primes conjecture states that there are infinitely many pairs of primes. The strong twin primes conjecture states that every prime \(p\) has a twin prime \((p+2)\text{,}\) although \((p+2)\) may not look prime at first. The tautological prime conjecture states that the tautological prime conjecture is true.
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