Came across some notes on curious properties of the number 17 in a blog, called Mathnexus. Then today one of my students announced that she was 17 years old... So I shared them with her and the class......and now with you:
It is the only known prime that is equal to the sum of digits of its cube (173 = 4913, and 4 + 9 + 1 + 3 = 17)
It is the only prime that is the average of two consecutive Fibonacci numbers. ... (Ok, that would be 13 and 21... now the only way Fibonacci numbers can have an integer value is if they are both odd [there are no consecutive even Fibonacci numbers}, and all the even Fibonacci numbers are twice the average of the two previous values... so there can be no Fibonacci number after 34 which is twice a prime) It is interesting that there are lots of odd prime Fibonacci numbers, 2, 3, 5, 13, 89, 233, 1597, 28657, for example , Sloane's A005478, and each of them has a prime index (except three, which is f4).
It is the least integer such that the sum of its digits in every base B = 2, 3, 4, 5, 6, 7, 8 is prime. (In base two ,17 is represented as 10001 and 1+0+0+0+1=2, a prime number; in base three it becomes 122, 1+2+2=5; in base four it is 41, in base five it is 32; in base six it is 25; in base seven it is 23; and in base eight it is 21... but in base nine it becomes 18, and 1+8 = 9 is not prime.) which makes me wonder... is there a prime number for which the sum of the digits in all bases two through ten is also prime?
There are exactly 17 ways to express 17 as the sum of one or more primes.(can you list them all?)
And just one more, there is no odd Fibonacci number that is divisible by 17. (Ok, how special IS that?... are there other (odd) numbers that do not divide evenly into any of the odd Fibonacci numbers?..YES, the smallest is nine. A good strategy for attacking this kind of problem is given in this blog by Tanya Khovanova. Tanya goes on to state that none of the odd Fibonacci numbers are divisible by 19, 23, or 27 also, so maybe this really isn't such an unusual event at all.)
17 is my favorite number...it's always seemed more elegant and unique than the rest...more of a feeling. Thank you for giving definition to my preference! This is great!
ReplyDeleteVery cool. On pi day at my school, anyone who can get *exactly* 17^2 digits of pi gets a whole pie, just like the person who gets the most digits. Not only is 17 so cool, but our school number is 34, which is twice 17.
ReplyDeleteI hate to be a fly in the ointment, but..."It is the least integer such that the sum of its digits in every base B = 2, 3, 4, 5, 6, 7, 8 is prime...in base eight it is 21" 21 isn't prime ^_^'
Paul, you are right, 21 is NOT prime..but if you read carefully, I said that the SUM of the digits is prime... and 2+1 = 3 IS prime... But thanks for reading and commenting.. I make enough mistakes that I definitly need the feedback.
ReplyDeletePat
In fact you make very few mistakes, as a teacher you know you always have 1-3 students each semester who like to point out when you're wrong during a lecture, which I'm sure is very rarely. And regarding mistakes, nobody's perfect, even Einstein made them. But Einstein admitted them! More important than making mistakes is admitting them, which probably explains why we're not in Politics. :-)
ReplyDeleteOK, John von Neumann only made one mistake, but it was a doozy. It held back research in Quantum Entanglement until John Stewart Bell came along 30 years later in the early 60's and discovered it, so the field begun by the famous EPR paper in '35 by Einstein, Poldolsky and Rosen could carry on.
I WILL point out one thing I wish you'd mentioned, and that was Gauss' accomplishment with the number 17. I'm not a teacher, just a fan, so I'll leave to the Teacher to remark on that one. Cheers.
Steven,
ReplyDeleteDid anyone ever tell you how much you resemble Teddy Roosevelt....
This comment has been removed by the author.
ReplyDeleteAnd now I'm a Sierpinski Tetrahedron on a dark green/light green checkerboard. How about that?
ReplyDeleteOK then back on topic. Gauss was able to solve a problem that had vexed generations, and he did it quite young. If I recall correctly it was to construct a 17-sided polygon, the hepta-something, using only a compass and ruler (and pencil of course). Is that right?