I am currently sidetracked to the Prime Curios website. It deals with primes, their properties, and some rather fascinating patterns in the numbers. It is basically a catalogue of the more interesting primes and the curious properties of individual primes.
Well, I decided to kick its tires, and see what patterns it could see in numbers that I clicked on. I chose their category of 12 to 15-digit prime numbers, since I thought they are probably going to come up with boring information. This proved wrong very early on. Each number had a link, and led to a page with a paragraph or two of information on it. I noticed that most, but not all numbers in the list were prime. But first some primes I clicked on:
100,123,456,789: Well, I clicked on the second number in the list before I noticed that the number contains all the digits from 0 to 9 in sequence, with 10 at the beginning. This is a pandigital prime number. In fact, it is difficult to dispute the site’s first assertion that it is the smallest prime to contain the string “0123456789”. But it is also the smallest prime where if you wedged a zero in between each digit, it would still be prime: 10000010203040506070809. This last number has a prime number of digits and a prime number of zeros.
100,529,784,361: also contains the numbers 0 to 9, with 10 at the start, but this time not in order. But when you break them into groups, you get perfect squares: ; and . Squaring each digit and concatenating the results gets us the string 100254814964169361, which is also prime.
101,111,111,111: I learned a new word today: emirp. An emirp is a prime spelled backwards. The definition is that the reversal of these digits yields a different prime. This is different from a number like “101”, which is a palidromic prime, since you get the same prime when writing the digits in reverse. Also the first two digits are “10” with 1’s repeated that many times.
1,919,110,119,191: This is a palindromic prime. You get the same number if you reversse the digits. Also the first six numbers: 191911 and the last six, 119191 are both prime and “emirp”. There is another oddity about 191911 and 119191: if you rotate the first of these numbers so that it appears upside down, you get 116161 (a prime); and the second one rotated, becomes 161611 (another prime, and emirp compared with 116161). Primes generated by such a rotation is called an invertible prime. It can only happen if a digit can be a recognizable digit when turned upside-down. Such primes must therefore only contain the digits: 0, 1, 6, 8 , or 9. 1,616,110,116,161, the rotation of the entire original number, is also prime.
6,886,699,889: This is a prime which is the same prime when rotated upside down. These invertible primes are referred to as Strobogrammatic primes.
28,116,440,335,967: This is the smallest multidigit prime number that is also a narcissistic number: the sum of each digit raised to the power of the number of digits, equals the number: .
Okay, fine. But what about the composite numbers listed by Prime Curios? Again, sticking to the same range of digits …
157,639,024,808: This composite number uses up all of the digits from 0 to 9 at least once. This makes the number pandigital, like others we’ve seen. It is also a perfect cube: . In addition, the number written backwards is prime.
2,504,730,781,961: This composite pandigital number is also the 61st Fibonacci number. 61 is prime, and are the last two digits of the number.
265,744,701,192,502: Someone with time on their hands figured out that if you multiply this number by 11 and add 1, you get a number which is a concatenation of the primes 2 to 29: 2,923,191,713,117,523, which can be read as the primes: 29, 23, 19, 17, 11, 7, 5, 2, and 3. They appear in reverse order of the primes except for the 3. However, this number is also composite.
For the sake of trivia, the largest known prime, according to this site, is: 282,589,933 – 1, a Mersenne prime containing 24,862,048 digits found in 2018 by Patrick Laroche of Ocala, Florida, running a GIMPS program on their computer.