Hits: 12
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 15digit 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 2digit 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:
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):



 17=17 (take care of the obvious)
 17=13+2+2
 17=11+3+3
 17=11+2+2+2
 17=7+7+3
 17=7+5+5
 17=7+5+3+2, the sum of the first 4 consecutive primes.
 17=7+3+3+2+2
 17=7+2+2+2+2+2
 17=5+3+3+3+3
 17=5+5+5+2
 17=5+5+3+2+2
 17=5+3+3+2+2+2
 17=5+2+2+2+2+2+2
 17=3+3+3+3+3+2
 17=3+3+3+2+2+2+2
 17=3+2+2+2+2+2+2+2


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


 17 is the only number, when cubed, its digits equal the number:
 17 = 2^{4} + 1, which makes it almost a Mersenne prime. Mersenne primes are primes of the form 2^{n} – 1.
 17^{2} = 289 can be expressed as the sum of 1, 2, 3, 4, 5, 6, 7, 8 distinct squares.
 There is one square, 17^{2} itself
 Sums of 289 in two distinct squares
 17^{2} = 15^{2} + 8^{2}
 17^{2} = 12^{2} + 12^{2} + 1^{2}
 Examples of obtaining 289 in 3 distinct squares:
 17^{2} = 16^{2} + 5^{2} + 2^{2} + 2^{2}
 17^{2} = 12^{2} + 8^{2} + 9^{2}
 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:
 2^{2} + 4^{2} + 10^{2} + 13^{2}
 2^{2} + 5^{2} + 8^{2} + 14^{2}
 2^{2} + 8^{2} + 10^{2} + 11^{2}
 3^{2} + 6^{2} + 10^{2} + 12^{2}
 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.
 17^{2} = 3^{4} – 4^{3}
 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”.
The reading of this table is, taking the first line:
1, 2, 3, 4, 5, 7, 8, 11 =>
289 = 1^{2} + 2^{2} + 3^{2} + 4^{2} + 5^{2} + 7^{2} + 8^{2} + 11^{2}
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