Science Forums 2

CraigD · 11-13-2008

Here’s a way to generate a starting value $A_1$ for Don’s generating formula

$A_n = \frac1{\frac{A_{n-1}}{\lfloor A_{n-1} \rfloor}-1}$

so that $\lfloor A_n \rfloor = p_n$ , where $p_n$ is the nth prime (tested for for the first 500 primes):

Begin with $n=1$ and the required inequality $p_n \le A_n < p_n +1$
Find algebraically $A_n = \frac1{\frac{A_{n-1}}{\lfloor A_{n-1} \rfloor}-1}$ , in terms of $A_1$
Solve the inequality $p_n \le A_n < p_n +1$ for $A_1$

Repeat steps 2 and 3 until you reach the desired $n$ .

For example:
$2 \le A_1 <3$

$A_2 = \frac1{\frac{A_1}{A_1 -2} -1} = \frac2{A_1 - 2}$

$3 \le \frac2{A_1-2} < 4$

$\frac{10}{4} < A_1 \le \frac{8}{3}$
…
$A_{10} = \frac{-192139677A_1+493134904}{200096592A_1-513556674}$

$\frac{15899835124}{6195037437} < A \le \frac{15386278450}{5994940845}$

... etc.

We can chose any value for $A_1$ that satisfies the greatest calculated inequality – choosing the equal value is a simple approach – and apparently be assured that we can calculate the primes up to $p_n$ using the generating formula.

To calculate the first 500 primes, $A_1=$
15073501454785746503327091792962580591389206596526 00133549958380812810366757065165351318863631121898 05088754668679404598405121140813129245153847730993 93116509224752615940520003349154516425731185540082 59720407821367807798119833292681432062301410370166 87595771322742390600315862122981380451503984941013 34800684765266878898731551080382057654332637777486 32943198726169060409838120619186737957385991864898 55274616712201322918797753182154374051901221643883 08401766200477671118778327976210659735624862328586 54963066780224690489741076518122543117075679553767 58134262077627499223084667550517509611291538649922 08460238041218167876258927758793873301156390329695 35787423752095874364099117082335959343236194781869 72886625439746358323543512860509390358963174892812 65616242291119899374599969237532278684445540152984 51998416667184097363645972478112430523253964734220 76268583872122010581309397796346550653512552762485 95775610026648609817260298625497878997388257097535 76389848126764707897188538012447534195114955790616 97513612836491589890200742469194942463622908742939 93685646066690793380712604662528277300752124615787 24024530729266658923876586891907943895406986125247 04527450013918513134098568308270445036996462945736 23401790800331999699278498314366466909118704288985 55504579399015828342165596477078211567396867561475 51693410894474070767164406309132429842346723176869 33263707673552995625299796709090068177685616423896 12184638886621898222564362672335971585149068968247 82659781043594073403139800623663367541651686138885 872797450033244899419
/
58730738457846712369826340648262300633444774983907 27638266470778615269971491665292322225550636323727 54943340292103730657241691107153212108006659110244 64444770069930762507049531752637095218702389563004 72671868034817523690954322826747978305232455713996 87221852110421144555772121627571238069919574312740 48312379712629723341277003483291278684987121090006 26392300980002192789352679079225669038538890282612 84250178662854328449539356908826372721147168559966 64330689080391442436234172363357128809712568640285 52418881599095254738336534436713440365888751446751 17634496607542124955096038497033007244704964526059 75105655789135940034413587214516108465042554163865 21028442559206881777089783965958129667817613706287 76604714941819486417598979802494861589879134463748 18647330634494353063371357000253993696192631497701 12032251322960920621433350293561696141471983174423 94244812297335112625826911481807488979992082072996 60744328690074587438204488162872696499342308524534 77143250845949228820778990534996717795255491921406 90007422880905267135023089484844501042549679306933 06291594975237393982723807460588492750996826237514 54177772504536730456144965905046511411264070667199 24718802040995924070029066445783627538937500713488 05373121101080095139634333461592440688781377704280 34253092196717448161422579637632820776337529951366 36204873845712523463654881671059209464838378854429 30354015090055056827775381445045083954590638547452 81683836154075418844349120501524322858819407545882 84285228929351709031168647854498330163790059744137 56096393667894862622

is precise enough. (Its denominator is 1520 digits)

Because each calculation requires only a few multiplications and additions, it's an easy calculation to make, much easier than the calculations to approximate the Sine and logarithms and iteratively approximate the zero of the function containing them.

Although I’ve not proven that this method will always be successful – its possible that some $A_n$ that satisfies $p_n \le A_n < p_n +1$ will not satisfy $p_{n-m} \le A_n < p_{n-m} +1$ – it appears to for all calculations I can make in a few minutes. More importantly, we haven’t proven that the $A_1 = x$ as given by

$\frac{\left[\left(sin(x^{(1/2)})\right)-1\right]^{-1}}{\left[\pi^2+\left[\ln\left(\ln\left(2\left[\left[2(x^{-1}+1)\right]^{-1}+1\right]\right)\right)\right]^{-1}\right]}-1 = 0$

If modest or someone else with a working program like the one in post #28 can modify it to use the new equation (it shouldn’t be hard to differentiate, needing only the product and chain rules, and some carefulness), it would be interesting to compare A few hundred digits of the estimated zero of $X$ to my big rational number above. To 500 decimals, it’s:
2.566543832171388844467529106332285751782972828702 31464596973352546639971989040034622398857147805665 89415300383386252694557180837585065234733899407590 15452147716305617441237846500920651165442820986967 99444086469195021290029958254446835359571462522431 94189226038317025371635511355609594950080639727211 11188080630943369037911871522603146919231148726991 01382281616159570290924835490077516263817781701705 01465893712305852748021584934680316196223087098420 52492295557540633289790051335145247812827882458860 36944358849212875826884884990827579513115666424648 20849280217151229993076859757596523704399063065354 07925624047164609395479942464328914535244340335467 28912555946828300675869093272900644507789827817806 46572326075380709000130766143755442519632323931974 44101894793461926400851780595643049017923189817237 13680529972307807980157357353519124741233224426245 55334814040204030157123671369216800571313500108714 69609483401152427491436846808849436797566037679245 00002211023112680763023278357128661735500471600507 58990823559294731332935283691934260732135205234475 64201678214095278196584532234664894564878811714234 31083061423838155882272075651801199499190609973138 44551046494747202015388384536230021753436402688469 88608135948517199422762601630425131670162358528085 11288133812294558351146855290775139229175383801288 73184842938429816881693161821371821961182096793893 94076251757447174244597019651368333949030078114849 02520373497194268565900019623252488180600825909134 66896412315136908706594026416435982690876451518198 999891129443265858404.

I’d be wonderstruck if they match, but have no clue why they should, other than Don’s “just so beautiful” argument.

From a practical computational perspective, like so many candidate prime number generators, this one’s not very useful, because in order to generate many primes, its $A_1$ must be very precise, and arithmetic on very precise numbers take a lot of computing time and CPU. From a math insight perspective, however, it’s wonderful if true, and a nifty (and maybe not easy) exercise to prove true or not.

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Hypography?

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