I Don't Hate Ulam Numbers Anymore

Ulam numbers were first described by Stanisław Ulam in 1964 in his book "Analogies Between Analogies."

Stan Ulam

In this book, Ulam proposed the sequence as a recreational math idea. The Ulam sequence begins with 

𝑈(1) =1 , U(2) = 2 and then each subsequent number is the smallest integer that is the sum of two distinct earlier Ulam numbers in exactly one way.  (not two ways, not zero ways, exactly one way.)

So, for example, the sequence starts:1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, …(yawn...)

I'm sure my first introduction to them was Martin Gardner when he  introduced them in the March 1966 issue of Scientific American in a column titled:"The Remarkable Lore of the Prime Numbers".  

As much as I loved the column, math, puzzles and problems, my Ulam Numbers vaccine didn't take.  

So Maybe you weren't around way back then, so here is a brief intro to Ulam Numbers, and along the way, the book that actually changed a 60 year dis-interest in them, Shyam Sunder Gupta's "Exploring the Beauty of Fascinating Numbers."  My first surprise; he had an entire chapter in this massive book on Ulam numbers.   Not a little chapter, but 18 pages????? Honestly I flipped by it for the first few weeks I owned the book.  He had a fascinating collection of such interesting topics and ideas,,,, 601 pages, and I was going from one beautiful idea to another... and then, I started reading the Ulam chapter, maybe it was a slow news day.

He starts out by introducing the definitions I gave above, and then the Ulam numbers up to 991

Now the parts I never got hooked up on is the numbers are interesting, And then the numbers that are missing, hmmm, now that's interesting.  I mean 23 is the first number that isn't there because there is no way in the numbers up to 18 to find two numbers that add up to 23.  You get so busy trying to eliminate the values that are the sum of more than one, you forgot the cardinal rule.... exactly one way.  And after waiting that long, the next one is 25....isn't there a quote about waiting for a bus and the two come along at once.  And Gupta interrupts the flow to point out some "Background and Known Results."  #2 was, Are there infinitely many numbers such as 23, 25, 33, 35, 43, 45...(he kept going but you know what came next, right....Oh no, 67, 92....What the heck....these guys are getting spooky... and really interesting.  

Gupta goes on with headings like, Density of Ulam sequences, non-ulam numbers, Odd and Even Ulam Numbers....(I quickly counted the odds in the first fifty Ulam numbers, but what I noticed was an early streak of 16, 18, 26, 28, 36, 38' six even numbers in a row, and even though the total ratio of odds to even was close enough to 50/50 that it made me wonder about the streaks,  continuing through I hit another streak of seven, all even, 502, 522, 524, 544, 546, 566, 568... seven evens, and the longest streak I found of odds was 3, until near the end of the ones he had listed, up to 991, there was a streak of six odds,  891, 893, 905, 927, 949, 983.  

Gupta goes on to sections on Prime Ulam numbers,,,,and now I'm thinking Primes that are Not Ulam numbers for having More than one way, or for not having any two number sums of Ulam numbers.  

He goes on to ways to generalize the U;am numbers to other sequences, creating a new version(s) of all the same things we have been going through.  It was like going into an ice cream store and they've created ten flavors you never heard of before...(personal note, I go into ice cream shock and fall back on my stand-bys, Vanilla, chocolate, or strawberry....except in Northern Michigan where it's always black Cherry.)

Keep in mind I'm telling you about only 18 pages of a 600+ page book, and I've been distracting myself for weeks with part of those other 500+ pages.  

If you like math, or if you have a child/friend/neighbor/spouse/other who likes math, this is a perfect gift for Birthdays, Christmas Anniversaries, Tuesdays...

I'll even put the Amazon Link here to make it easy  This gem is already on the best seller lists in several countries, (and I'm thinking here, now or soon.    Enjoy