Prime Curios III: The number 23

Number23.jpg
“The Number 23”, a 2007 movie which starred Jim Carrey and was directed by Joel Schumacher.

23 was the number featured in two movies. The 1999 movie called “23” was in German,  starring August Diehl, Fabian Busch, Dieter Landuris, Jan Gregor Kremp. The Hollywood movie called “The Number 23” was released 8 years later, starring Jim Carrey, Virginia Madsen, Logan Lerman, and Danny Huston. Both movies appear to be psychodramas based in some way upon the fear of the number 23, a condition known as Eikositriophobia. The phrase “the 23 enigma” figures in this film, usually uttered by people who find a mystical significance with the number 23.

Yes indeed there are people in the world that fear a specific number of some kind. Each number has its own long and nearly unpronouncable name. The most common fear would probably be Triskaidekaphobia, the fear of the number 13 (oh yeah, another prime number). But let’s stick to the number 23 for this article.

    • It is part of the expression “23 skidoo!”
    • 23 is a number whose digits add to 5, the next prime.
    • 23 is a part of the Birthday Paradox: 23 is the minimum number of people to have in a room, where the probability of two people having the same birthday is just over 50%. Related to this, is the question of how many ways can 23 people be paired? It turns out that this becomes {23 \choose 2} = 253 \approx 365\ln{2}.
    • Twin primes are a pair of prime numbers which are also two consecutive odd numbers. Examples are 3 and 5; 5 and 7; 11 and 13; 17 and 19. The first prime that cannot be “twinned” in this manner is the number 23.
    • 23 is also called a “factorial prime”, because the next whole number, 24, is a factorial: 24 = 4\times 3\times 2\times 1 =4!.
    • Reversing the digits makes it a power of 2: 32 = 2^5=2^{2+3}.
    • e^\pi, rounded down to the next lower integer, is 23, or: \lfloor e^\pi \rfloor =23.
    • The sum of the first 23 primes is 874, a number divisible by 23.
    • 23 is made from digits which are the first two primes, 2 and 3. Wikipedia calls a number like 23 a Samarandache-Wellin prime, where putting together the digits of the first n consecutive primes makes the next Samarandache-Wellin prime. “2” is the first prime, so is the “sum” of only one prime. The second prime has to be the concatentation of the first two primes, so “23” is the next one. 5 divides “235”, so that doesn’t work; but “2357” is prime, the concatenation of the first 4 primes to make the third Samarandache-Wellin prime.
    • 23 is also the number of chromosomes in a human; the atomic mass of sodium; the degrees of tilt of the Earth’s axis; the Bible’s 23rd Psalm; and the number of times Roman emperor Julius Caesar was stabbed in the Theatre of Pompeii.

Prime Curios II (the number 17)

The prime curios website has given me more to discuss so that I think I will choose some other numbers. In the first post in this series, I chose high, 12 to 15-digit prime numbers to discuss, and it wasn’t hard to find discussable numbers. Nearly anything I clicked on had something interesting about it.

Now I am going back to 2-digit numbers. The first one I chose is 17. Apparently, there are 17 ways to write the number 17 as the sum of primes. You are encouraged to try this out yourself before revealing what is underneath the “spoiler” below:

Click on this spoiler to see the 17 ways...

We have at our disposal 2, 3, 5, 7, 11, 13, and 17 to work with, so let’s see how we can do this (the last few took awhile):

        1. 17=17 (take care of the obvious)
        2. 17=13+2+2
        3. 17=11+3+3
        4. 17=11+2+2+2
        5. 17=7+7+3
        6. 17=7+5+5
        7. 17=7+5+3+2, the sum of the first 4 consecutive primes.
        8. 17=7+3+3+2+2
        9. 17=7+2+2+2+2+2
        10. 17=5+3+3+3+3
        11. 17=5+5+5+2
        12. 17=5+5+3+2+2
        13. 17=5+3+3+2+2+2
        14. 17=5+2+2+2+2+2+2
        15. 17=3+3+3+3+3+2
        16. 17=3+3+3+2+2+2+2
        17. 17=3+2+2+2+2+2+2+2

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Some factoids about the number 17: Did’ja know …

      • 17 is the only number, when cubed, its digits equal the number: 17^3 = 4913 \Rightarrow 4+9+1+3=17
      • 17 = 24 + 1, which makes it almost a Mersenne prime. Mersenne primes are primes of the form 2n – 1.
      • 172 = 289 can be expressed as the sum of 1, 2, 3, 4, 5, 6, 7, 8 distinct squares.
        • There is one square, 172 itself
        • Sums of 289 in two distinct squares
          • 172 = 152 + 82
          • 172 = 122 + 122 + 12
        • Examples of obtaining 289 in 3 distinct squares:
          • 172 = 162 + 52 + 22 + 22
          • 172 = 122 + 82 + 92
          • There are several more with 3 distinct squares, when you get to 4 or more terms.
        • Obtaining 289 from 4 distinct squares, each appearing once:
          • 22 + 42 + 102 + 132
          • 22 + 52 + 82 + 142
          • 22 + 82 + 102 + 112
          • 32 + 62 + 102 + 122
        • Of course there are more, and click on the spoiler below at the bottom of this article to get a full list of the ways ot obtain a sum of 289 under these conditions, after trying at least a few yourself.
      • 172 = 34 – 43
      • 2(17+17) has “1717” as the first four digits. (17,179,869,184)
      • According to MIT hacker’s lore, 17 is referred to as the “least random” number.

The last point is special. Why is 17 considered “least random”? We have to not consider computers or shuffling of cards or mixing tickets in a hat, here. This is a purely human phenomenon. If a thousand people were asked to “radomly” speak out a number between 1 and 20, one would expect that each of the 20 numbers would be equally likely to be spoken out. This is far from the case. Instead, one would find that 17 is spoken out disproportionately more often compared with the other 19 numbers. If people were asked to “randomly” choose a number, a surprising number of them are primes, and 17 is the prime chosen most often. This is why it is said that 17 is the least random number to choose. The reason we choose prime numbers most often has less to do with our affinity for prime numbers and more to do with the fact that we seem to conflate “strange” or “unique” with “random”.

Click here to see 15 ways to get 289 with up to 8 unique perfect square terms.

The reading of this table is, taking the first line:

1, 2, 3, 4, 5, 7, 8, 11 =>
289 = 12 + 22 + 32 + 42 + 52 + 72 + 82 + 112

1, 2, 3, 4, 5, 7, 8, 11
1, 2, 3, 5, 9, 13
1, 3, 4, 5, 6, 9, 11
1, 3, 5, 6, 7, 13
2, 3, 4, 8, 14
2, 3, 5, 7, 9, 11
2, 4, 5, 6, 8, 12
2, 4, 5, 10, 12
2, 4, 6, 8, 13
2, 4, 10, 13
2, 5, 8, 14
2, 8, 10, 11
3, 6, 10, 12
8, 9, 12
8, 15
17

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The Prime Curios Website

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: 100=10^2; 529=23^2; 784=28^2; and 361=19^2. 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: 2^{14}+8^{14}+1^{14}+1^{14}+6^{14}+4^{14}+4^{14}+0^{14}+3^{14}+3^{14}+5^{14}+9^{14}+6^{14}+7^{14}  = 28,116,440,335,967.

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: 157,639,024,808 = 5402^3. 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.