# Only The Lonely These primes are just ‘Equally Lonely’

Just to update: Prime numbers are only divisible by 1 and themselves. They play a major role in discrete mathematics, perhaps the most elegant branch of mathematics. But they also do their work collectively in many current technologies, such as cryptography. Exact prime numbers are prime numbers that are no longer prime by switching any number. So they are basic, but at the same time “weak”. After all, any misspelling, in and to any number, makes it non-primary. He listens closely.

Take, for example, the number 294.001. Exact prime number? Yes, because any bold typo in the string below is “fatal” and renders it non-primary:

094001, 194001, 394001, 494001, 594001, 694001, 794001, 894001, 994001

204001, 214001, 224001, 234001, 244001, 254001, 264001, 274001, 284001

290001, 291001, 292001, 293001, 295001, 296001, 297001, 298001, 299001

294101, 294201, 294301, 294401, 294501, 294601, 294701, 294801, 294901

294011, 294021, 294031, 294041, 294051, 294061, 294071, 294081, 294091

294000, 294002, 294003, 294004, 294005, 294006, 294007, 294008, 294009

Are these prime numbers rare? They are certainly much rarer than the “regular” varieties. But all things considered, it’s okay. 294,001 is the smallest, followed by 505,447, 584,141, 604,171 and 971,767. And over a million there are many, many without limits in reality. By comparison: among the ordinary primes we already have 25 smaller than a hundred, and 78,498 less than a million. So it’s a little thinner, those exact numbers How about those broad ones?

You can type any number with or without leading zeros. 294001 or 0294001 or 00294001: anything is possible. At the end of 2020, it has been proven that primes also exist with this special property: no matter how many zeros you put in front of them, they remain sensitive. Punishment right? Small details: none found yet. But it is proved that there are prime numbers that you can put zeros in front of them infinitely and with which you may not write a single number wrong, or become non-prime.

What about the unit now? With his bestselling book, The Loneliness of Primes, Italian novelist Paolo Giordano shows that the general public is indeed interested in primes. And the solitary aspect of these special quantities does not leave us untouched as human beings. In his novel, Giordano talks about prime twins, and how they explain loneliness in prime numbers: “Lonely and lost, close to each other, but not close enough to actually touch.”

What Giordano writes is indeed true. If you climb higher in the tower of primes, you will never find primes on top of each other. They are always separated by at least one floor between them. They are then called double primes, like 3 and 5 or 5 and 7. And make no mistake, 2 and 3 touch each other, and (3, 5, 7) even a triple prime. But that’s it. Because except for 2 and 3, there are no primes touching each other, and with the exception of (3, 5, 7) there are no primes. So he feels very lonely on this sign.

Now for the good news for 2021: Large-scale primes, while rare, are actually no more isolated than the common ones. Because they come in a complete series of two, three, or even an unlimited number of consecutive primes. Not necessarily as double primes, but they are still close in prime sequence. So there are two sequences of very precise prime numbers. As well as groups of three. and so on and so on. In general, these numbers aren’t any more isolated than regular primes, they’re just kind of lonely and missing, close to each other, but not close enough to actually touch.

There is more good news. If this mere mortal is correct, the pretty geeks behind this discovery still need work. Find the first wide precision prime number, for example, and find out exactly what it is. Furthermore, they might wonder if large-scale twin primes also exist, and whether there are many of them? We can live with these doubts for now, because we know so little about double primes anyway. Except that they are alone. Mathematicians believe that there is an infinite number, but this is not certain. Maybe something for 2022?

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