And One for a Special Birthday, Why 17 is Special

  

One from the archives, reminded me my grand-daughter is turning 17 then. Happy Birthday, Zoe!


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?)

With any number of points less than 17, it is possible to color all the segments between them with two colors without any triangle having all three sides of the same color. With 17 or more, it is not possible.

The sum of the squares of the first seven primes (all primes up to 17) is 666, the "Number of the Beast." 22+32+52+72+112+132+172=666

And 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 mph is an unusual speed limit, but on the campus of Hampshire College in Amherst all the speed limit signs have been changed from 15 to 17 miles per hour to honor retired mathematics professor David Kelly. Kelly spent his career fascinated by the number 17. There is at least two others in the US, at Mountain View, California and Fiesta Mall in Mesa, Az.

In 1796 (don't miss the 17) Gauss proved the Heptadecagon (17-gon) was constructible with straight edge and compass.  This was the first progress in polygon constructions in 2000 years  (This statement obviously excludes other methods of polygon construction over the years including origami, ).  

Gauss was so pleased that he wanted to have a 17-gon on his tomb, but when the mason tried to msake one it kept looking like a circle so he opted for a 17 pointed star.  

A 1802,few years after his construction, Gauss showed that if a regular polygon has a number of sides that is a power of two, times one or more distinct Fermat Primes ( 3, 5, 17, 257, 65537) then it is constructable.  Ge stated without proof that it was also necessary.  The first rigorous proof of necessity was by Pierre L. Wantzel in 1837 (often cited as the Gauss–Wantzel theorem). Wantzel showed by analyzing cyclotomic equations.  

And a hat-tip to Steve Colyer who reminded me years ago I had omitted this key 17 oddity.