The sequence of Mersenne numbers (which are those of the form 2ⁿ – 1), which we talked about last week, includes two primes, 3 and 7, among its first few terms, which might lead us to think that it is not they will be scarce. Nothing more fake. Among the numbers less than 100, we only find one other prime, 31, which is equal to 2⁵– 1, and among the three-digit numbers there is only one, 127 = 2⁷ – 1. These four primes, 3, 7, 31 and 127, were already known in Euclid’s time, but Mersenne’s fifth prime was not discovered until the 15th century: 8191 = 2¹³ – 1. In the 16th century, the Bolognese mathematician Pietro Antonio Cataldi discovered two more: 131071 = 2¹⁷ – 1 and 524287 = 2¹⁹ – 1, and it would take until the eighteenth century for Euler, applying Cataldi’s tentative division method, to discover the eighth Mersenne prime: 2147483647 = 2³¹.
Mersenne’s eighth cousin is peculiar enough for us to dedicate a few lines to it. First of all, it is a double Mersenne number, so named because 31, the exponent of 2, is itself a Mersenne number. There are other double Mersenne numbers, and I’m sure my astute readers can find some of them (and I’m also sure they can’t find them all, unless they consult a specialized page).
But the number 2147483647 is also very popular in the computer world, as it is the largest integer in systems with 32-bit architecture (why?), and for this reason it is also the maximum possible score in some video games. As a curiosity, in the form (214) 748-3647 is the sequence of digits that is most frequently presented as an example of a telephone number in the United States.
From the eighth, the Mersenne primes grow dizzyingly and are further and further apart from each other, until reaching M₅₁, which, as we saw last week, has almost 25 million digits and is the largest known prime number so far.
Distance between consecutive primes
But the enormous distance between two Mersenne great primes is not the greatest possible, since we can find consecutive primes as far apart as we want.
There are only two consecutive primes in the strict sense: 2 and 3, since 2 is the only even prime. Primes that are odd in a row, like 3 and 5, 11 and 13, 17 and 19, 29 and 31, are called twins (can you find other pairs of twins?). In this sense, 3, 5 and 7 are “triplets”, the only ones, by the way (why?).
Among the first 100 natural numbers, the furthest primes are 89 and 97, which means that in this interval there is no sequence of more than seven consecutive composite numbers. How would you find (without consulting any table, of course) a sequence of eight consecutive composite numbers?
You can follow MATTER in Facebook, Twitter and Instagramor sign up here to receive our weekly newsletter.
#Distant #cousins