The sphenic numbers we covered last week all have eight and only eight divisors since as a product of three different primes they have the form n = pqr, so their divisors will be in addition to 1 and the number itself, the three prime factors plus the three binary combinations of these prime numbers, i.e.: 1, p, q, r, pq, pr, qr, n.
As we have seen, there can be two or even three consecutive sphenic numbers; but it cannot be four, since every fourth consecutive number is a multiple of 4 and therefore contains the repeated factor 2, and a sphenic number is by definition the product of three different prime numbers.
Goldbach’s conjecture
It seems that any even number greater than 2 can be expressed as the sum of two (not necessarily different) prime numbers:
4 = 2+2, 6 = 3+3, 8 = 3+5, 10 = 3+7 = 5+5, 12 = 5+7, 14 = 3+11 = 7+7…
If we take the sequence of prime numbers (2, 3, 5, 7, 11, 13, 17…) and pair them in all possible ways in order, with themselves and with the others (and the 2 after usage disregard). to get 4), by adding the two members of each pair, we get the consecutive even numbers. And since the larger a number is, the more different ways there are to express it as the sum of two others, it seems clear that we can always split an even number into two odd terms, which are also prime numbers.
It seems clear, and most mathematicians are convinced of it; but so far no one has been able to prove it. It all started when the Prussian mathematician Christian Goldbach wrote Euler a letter in 1742, urging him to find a proof for this very reasonable, almost self-evident assumption, which has since come to be known as Goldbach’s conjecture. But the great Euler, who never resisted a problem of numbers, was unable to find a proof, and all who have tried after him, who have been legion, have failed.
Computer brute force has shown that Goldbach’s conjecture holds for all even numbers under a hundred trillion (a 1 followed by twenty zeros); but that’s nothing compared to infinity, and while most mathematicians believe the conjecture is true, some believe that very large primes might surprise us more than once. And on the other hand, nobody could find a counterexample, that is, an even number that cannot be expressed as the sum of two prime numbers, which would show that the conjecture is wrong. A seemingly trivial problem within the grasp of a child that has proven to be one of the most difficult in the history of mathematics (the most difficult, according to some).
As we saw last week, in 1973 the Chinese mathematician Chen Jingrun used puzzle theory to prove his eponymous theorem, according to which any sufficiently large even number can be expressed as the sum of two primes, or one prime and one semiprime (Remember that a half prime is the product of two prime numbers that are not necessarily different, such as 9 = 3×3 or 77 = 7×11), which could be a first step in proving Goldbach’s conjecture. Now all that is left to do is follow the steps below.
And finally, and fitting to the topic, the most difficult meta-problem:
How is this part of The Science Game different from all previous ones?
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