FEREGO Applied Math

Fiction warning. This story is fiction, but the mathematical and scientific aspects are accurate.

Introduction

This web page illustrates some of our mathematical abilities, and partially serves advertising purposes for FEREGO. FEREGO offers applied mathematics consulting services (in addition to computer consulting services).

This web page is not implied to represent FEREGO's abilities.

Bob's neat numbers

All names are ficticious.

I don't know about you, but I think the Bob numbers are kind of neat. These are the numbers that begin with a 1, then a 3, and after that the next number is the sum of the previous two numbers. Of course the next number is 4, and after that 7, and so on. Bob discovered that this series of numbers has some very interesting properties. Needless to say, these numbers get large very quickly. Even though they add, they multiply quickly, like rabbits.

Mathematically, this can be stated as

x_n = x_n-1 + x_n-2

where the sequence starts out at 1 and 3.

Bob and Fibonacci are related, of course, except that Bob is not Italian, and Fibonacci is not a rabbit. So who's this rabbit Bob? Well, ok, so I met him while driving to Yosemite, and we had a nice talk, and he told the other rabbits. So you might wonder "where is this Bob fellow?" Bob lives right next to Swen and Oliv, two very nice, and very tall (and very wise) old trees. They stand by the road all day long, and presumably all night long.

And Bob told me about these neat numbers that he found while he was out looking for food in the desert. Sometimes, he said, acorns are piled up in this neat order, and he started counting, but as a rabbit he never got past 7881196, the 32^nd Bob number. Even though he could not count high, he was still thankful for the discovery he found in nature. But Bob had another friend rabbit who was more educated, and who knew lots more big numbers, and she, Kathy rabbit, said that she had gotten up to the 3003^rd. That's a big number he thought, and he wondered if there was anything larger.

He said that Kathy rabbit had helped him to understand that this is a simple example of an iterated algorithm, and the ratio of two successive elements in this sequence exhibits a non-chaotic behavior, and converges to a constant. The other animals were amazed. I asked Bob in reply how this could be, and he thought about it for a while and he said that he did not know of any little animals that new the answer, and so he thought, and he said he would try to ask the bear, the "big bear" (known as B. B., for short, as in B. B. King, the King bear, at least so as they say in "the jungle"). But Bob is afraid of B. B. of the jungle, so he thought he would ask Kathy rabbit instead.

Kathy rabbit told him that this sequence is naturally amenable to machine computation, and that she knows lots of computer nerds who could help out with this difficult problem in function iteration. Bob asked what a nerd was, and Kathy rabbit said that its like a special kind of animal -- you can tell by the tennis shoes, sometimes worn in bed. And so Kathy rabbit asked a friend to help with the so called "numbers of Bob," the rabbit, as they came to be known in the forest. Kathy rabbit thought a while about who she might ask, since she knows lots of nerds. She thought, "I'll ask B. B." She knows B. B., who has a huge cave, but who every little animal is afraid of. And so B. B. the King bear of the jungle helped Kathy the rabbit who helped Bob. And Bob was happy now, thanks to all the animals of the forest, because now he was finally able to get past the 32^nd Bob number.

And this gave Bob more time to climb around in Swen and Oliv, his two favorite trees. He thought "trees are nice."

Here are some of the "numbers of Bob" ...

C:\>1.6 | more bob(3)= 4 bob(4)= 7 bob(5)= 11 bob(6)= 18 bob(7)= 29 bob(8)= 47 bob(9)= 76 bob(10)= 123 bob(11)= 199 bob(12)= 322 bob(13)= 521 bob(14)= 843 bob(15)= 1364 bob(16)= 2207 bob(17)= 3571 bob(18)= 5778 bob(19)= 9349 bob(20)= 15127 bob(21)= 24476 bob(22)= 39603 bob(23)= 64079 bob(24)= 103682 bob(25)= 167761 bob(26)= 271443 bob(27)= 439204 bob(28)= 710647 bob(29)= 1149851 bob(30)= 1860498 bob(31)= 3010349 bob(32)= 4870847 bob(33)= 7881196 bob(34)= 12752043 bob(35)= 20633239

and finally, grunt, and with B. B.'s help, Bob computed the 5000^th Bob number, which is

C:\>1.6 5000 bob(5000)= 867363714658958853836858990837346279887492969082666977189105 168296032437457928943194094450659340183118067592764081066578 781405815712523222905923521828164543178306428062949156965072 596007824286305795272002893999008902437989509053981977793368 494102290220756352611289478561786225123836516119871710645821 997853641446618928553933299576550112936216927617570489075813 808350728627765184738385760898879117903858039941429947874039 539633004621357762641010286712142220431005956697065037124222 603215919383497418339098105605319178446629636083860155352921 143942735485495877545317130353296098698397464683112864659121 507658836078297513391129277284605454813209544392610850636568 570658101962652872653748071140650434941658734777448207370077 434699146248491548582142221213015365928816562167262238774979 750201386034241421924145732399474311930142111947690289516257 853684573558975072156722157563570859476242996732150137092238 341198859380870743351107200376928897247896452942477605132937 872537489692053931374673385300829341878493809691291010139586 4289472119273964080078127

Math

Years ago, we accidentally computed the 5000^th power of the number (1 + Sqrt[5])/2, and we found, purely by accident, that to 2150 decimal places it had the value ...

8.673637146589588538368589908373462798874929690826669771891051682960324374\ 579289431940944506593401831180675927640810665787814058157125232229059235218281\ 645431783064280629491569650725960078242863057952720028939990089024379895090539\ 819777933684941022902207563526112894785617862251238365161198717106458219978536\ 414466189285539332995765501129362169276175704890758138083507286277651847383857\ 608988791179038580399414299478740395396330046213577626410102867121422204310059\ 566970650371242226032159193834974183390981056053191784466296360838601553529211\ 439427354854958775453171303532960986983974646831128646591215076588360782975133\ 911292772846054548132095443926108506365685706581019626528726537480711406504349\ 416587347774482073700774346991462484915485821422212130153659288165621672622387\ 749797502013860342414219241457323994743119301421119476902895162578536845735589\ 750721567221575635708594762429967321501370922383411988593808707433511072003769\ 288972478964529424776051329378725374896920539313746733853008293418784938096912\ 910101395864289472119273964080078126999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999999999999999999999999\ 999999999999999999999999999999999999999999999999999999999999999999884708112283\ 300578777133501760846871339992779819786390758985910172295199 *10 ^1044

So what does this have to do with math?

We became very concerned about the difference between this number and the 5000^th Bob number.

Finance

Are there any applications in finance? (See above). Some high accuracy ability is integrated into some of our financial tools, although practical applications may be far off.