_GetPrime() returns larger primes, 300 digits+

BinaryBrother · May 25, 2015

This function returns 300+ digit primes, by pulling them from http://primes.utm.edu. The site that I'm using gets pretty sporadic with the content, but this is what I came up with.

I couldn't find much assistance for generating primes this large. I'm working on getting a complete mirror of the database, so I can implement a local function that doesn't require internet access. I somewhat like the idea of a cached approach, rather than actually finding a prime this large, as it would be very fast.

#include-once
#include <Array.au3>

Func _GetPrime($pStart, $pStop = 1000)
    Local $lPrime_Index, $lPrime_Pointers[1], $lPrime_Source, $lPrime_Dirty[1], $lPrime_Clean
    Local $lPrimes_Array[1]
    $lPrime_Index = BinaryToString(InetRead("http://primes.utm.edu/curios/index.php?start=" & $pStart & "&stop=" & $pStop, 1))
    $lPrime_Pointers = StringRegExp($lPrime_Index, '(page\.php\?number_id=\d{1,10})\" title\=\"prime\"\>', 3)
    If IsArray($lPrime_Pointers) Then
        For $N = 0 To UBound($lPrime_Pointers) - 1
            $lPrime_Source = BinaryToString(InetRead("http://primes.utm.edu/curios/" & $lPrime_Pointers[$N] & "&showall=Show+all+digits%3F", 1))
            $lPrime_Dirty = StringRegExp($lPrime_Source, "\<code\>(?s)(.+)\<\/code>", 1)
            If IsArray($lPrime_Dirty) Then
                $lPrime_Clean = StringRegExpReplace($lPrime_Dirty[0], "(\&nbsp\;|\<br\>|\s)", "")
                _ArrayAdd($lPrimes_Array, $lPrime_Clean)
                $lPrimes_Array[0] = UBound($lPrimes_Array) - 1
            EndIf
        Next
        Return $lPrimes_Array[Random(1, $lPrimes_Array[0], 1)]
    EndIf
EndFunc   ;==>_GetPrime

Use:

$RandomPrime = _GetPrime(350,500); Random prime, between 350 and 500 digits.

Here is the 'cached' prototype that I'm working on...

Func _GetPrimeLocal($pMin, $pMax = 1000)
    Local $lUsable_Primes[1], $lPrimes_Cache
    $lCache = "2357111317192329313741434753596167717379838997101103107109113127131137139149151157163167173179181191193197199211223227229233239241251257263269271277281283293307311313317331337347349353359367373379383389397401409419421431433439443449457461463467479487491499503509521523541547557563569571577587593599601607613617619631641643647653659661673677683691701709719,7891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891234567891,718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427427466391932003059921817413596629043572900334295260595630738132328627943490763233829880753195251019011573834187930702154089149934884167509244761460668082264800168477411853742345442437107539077744992069551702761838606261331384583000752044933826560297606737113,23232323232323232323232323232323232323232323231919191919191919191919191919191919191917171717171717171717171717171717171313131313131313131313131311111111111111111111117777777555553335555577777771111111111111111111111131313131313131313131313131717171717171717171717171717171717191919191919191919191919191919191919192323232323232323232323232323232323232323232323,1234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567,3357919051293446684817338475525336738410860683671992917156014373893439459981205349461903840857391256114552436727030705242812636090289386090253261128312388690032525006703900566545832881453504420702454161952515801782756924440723510485553617697701860843302292487377415029400870876382968571950877132931544109243793366659520676094343242287408182490885436245732490266782865577,147905612304049314530022347858051795393169843567686496119778103525328675398119462130526918624189025093221823879535477996560410449665190880468513606241420472470681055332721887591747863405964601930591007609161946038677991696595604464433826327996801243951556854274701334312260550305897135170116383049658549024443746904511563699126930309386309517504941636470564772320067276393,11243941652563674986498110100111211214413169141961522516256172891832419361204002144122484235292457625625266762772928784298413090031961321024331089341156351225361296371369381444391521401600411681421764431849441936452025462116472209482304492401502500512601522704532809542916553025563136573249583364593481603600613721623844633969644096654225664356674489684624694761704900715041725184735329,315791951375393537137595555337555955191395351995195751755791151795317131135377351919777977373317997317733397199751739199735799971153399111973979977771537137371797357935195531355957399953977139577337393111951779135151171355371173379337573915193973715113971779315731713793579595533511197399993313719939759551175175337795317333957313779755351991151933337157555517575773115995775199513553337335137111,24467476921682913438488192789239875454052536352549377837017690003767392142115603881393841041310716327816213613354449520915894649236940295705032203772143700063945600771074430370325781739206543222519232991385636612822084739163686757649627381681950058593932837672011365718824444153413729381893208613692665940794284282462079335312142411834200416495641677380129898882762716949168399500016681107627423520001,19999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999" & _
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            "11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111,1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111112887,1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111151191,1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111114144411,1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111114111111111" & _
            "111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111,1313879718456104225921995528497555855625152414017051819746489965175418594035237238181967239932841927943368022817082758088019245450978689926961358127631787144754373062120106902504163460853000843067796699510293924385333858122456855065858902248896798237625003273316840048212276610565777594650220087971045508055611314300070786176626378923084556948528977153135635206046615173944871186308702998089733796437598174864856900497873863589167218546523213052953471290292740839713119247294213720517263787354623928731924441813068650485005632246166869508621489664962699281739398162579362377493635567561343072827446535980932480283909225368443393486477711655376434350643031227524710815041165924570415486219444700695031601993251625411314037854648482400238147123249264167061961509379444920959953800364778020888142773659393688842351560111550969115614788364312559110854239694359378947557812483113318119934088060561737965196192041096299817733313038159648080136912212919336690293080257579650807503654066749005924284436981469,1332061140839377881275519727319087664563488856868363921090395424176003152616860694181381143398958886259607293657001126184200768318416415606007833158939494315794328961487646727235838064540020547096394715096987075662967270767111833128464892735927736680676281574819109293384227032660487105717000617743694091166592360430209859676725860069486092083162245353787489347996458344474036872405942603186910010107030249504890805053796812313931063356431538211577486558808551432480653655410786671149204990680160299130601476673122202346746073152960215724110679148813387815550919294278374132793382003237507202781781016933560748035199284389762236567868983336572644092830076478031437341713748965371275907861617842781207938340308618406806132341761652317375596844412105643666482330760029638578300024864280034704974456560072577302079710967422894034833870079056341042313954950303306609775366758875654233649026454012791177182484068909800375740635567792618409877923903328804607686747508551182788215187315706825259409787659457,1717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171717171738858369,222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222" & _
            "2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222252273,2222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222224489,2372745398987024026246661730569400071940852598530913706750961313093613019661136660043304214953229588549115824235879381842077106653008440157763482515826755896463281987961979543338530673344287018463219016115004917088496306089189981437869746067778819705086659226608214905773265406521005821190696164966769062789270607392466810411611364927251425724094853199855468930458472451779485739850642137721909433918071274371636467786383202340657583081021199102890278550492075053970663402100153191893555813143629199328494500822016926378226575514315855644658990142857007620141512167568953260848559217323755974661494616621625481469726141488042628605904007828553835113010863602272246388549781675554738720289081288962182158908888074895912630253490848795442738869861959492164311090600752598942985608896520433889214529842303378943565378331077082821046504392936708043242708752032968977683539225351550077540671477173072122185514120758317186048432316935013230715793661245186916334985219768354365429487437645709510479749390883,3000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066833,32222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222" & _
            "22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222233333255227,3333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333323333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333,3333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333335775533,3333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333533753553373,4000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000" & _
            "000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000609469,4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444446969649,6000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000869899,9444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444448484498999,999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999" & _
            "9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999499999,"
    $lCache_Array = StringSplit($lCache, ",")
    For $N = 1 To $lCache_Array[0]
        If StringLen($lCache_Array[$N]) >= $pMin And StringLen($lCache_Array[$N] <= $pMax) Then
            _ArrayAdd($lUsable_Primes, $lCache_Array[$N])
            $lUsable_Primes[0] = Ubound($lUsable_Primes)-1
        EndIf
    Next
    Return $lUsable_Primes[Random(1,$lUsable_Primes[0],1)]
EndFunc   ;==>_GetPrimeLocal

Edited May 25, 2015 by BinaryBrother

jchd · May 25, 2015

This function returns 300+ digit primes, by pulling them from http://primes.utm.edu. The site that I'm using gets pretty sporadic with the content, but this is what I came up with.
I couldn't find much assistance for generating primes this large. I'm working on getting a complete mirror of the database, so I can implement a local function that doesn't require internet access. I somewhat like the idea of a cached approach, rather than actually finding a prime this large, as it would be very fast.

I just can't resist replying with a very old (watch the date!), definitely hilarious, answer posted in a newsgroup about the number of large primes in a given range (at the time it was cutting-edge). Please realize that this was talking about 512-bit primes, while you're talking about 300+ digits.To place the explanation in an understandable context, let's say that a publicized composite, RSA-2048(*) (previously part of the RSA challenge) is a 2048-bit number, product of two equally (up to something) large primes, each in the range of 1024-bit, that is, each about 308 digits. So adjust the figures below accordingly!

Date: Thu, 20 Jul 1995 13:05:02 -0400
From: bostic@CS.Berkeley.EDU (Keith Bostic)
Subject: Using the advanced storage technology available to the NSA ...
To: /dev/null@python.bostic.com

Forwarded-by: matthew green <Matthew.Green@fulcrum.com.au>

From: hansm@wsinti05.win.tue.nl (Hans Mulder)

>> But cracking RSA is easier than running a sieve until you have all primes
>> until those used in RSA.  Besides that, you would not have enough storage
>> for all those primes.

> Good point :) Forgot about that little space problem (oops).  Of course,
> who can really say other than the NSA?

Astronomers should know about it.  RSA is typically performed using 512
bit prime numbers.  There are approximately 3.778e151 such prime numbers.
Using the advanced storage technology available to the NSA, it should be
possible to store a 512 bit number in a single hydrogen atom.  A typical
universe (e.g. ours) contains approximately 1e90 hydrogen atoms.  If the
NSA has hidden 3.778e61 universes in an inconspicious little building in
Maryland, astronomers should notice some deviations in the gravity field
in the area.

-- 
Hope this helps,

HansM

(*) For the record, RSA-2048 = 25195908475657893494027183240048398571429282126204032027777137836043662020707595556264018525880784406918290641249515082189298559149176184502808489120072844992687392807287776735971418347270261896375014971824691165077613379859095700097330459748808428401797429100642458691817195118746121515172654632282216869987549182422433637259085141865462043576798423387184774447920739934236584823824281198163815010674810451660377306056201619676256133844143603833904414952634432190114657544454178424020924616515723350778707749817125772467962926386356373289912154831438167899885040445364023527381951378636564391212010397122822120720357

Seriously, pick any randomized (whatever method you whish) integer in the 300+-digit range, call it n, then ask Wolfram website about NextPrime[n]. You'll get a 300+-digit prime in no time.

Here's a Mathematica v9.01 sample run (I typed 300 digits like mad on the keypad):

In[1]:= n=1654486103240894053403206870530135065304612049816540984530749803013049804150490864068468743049435046830646806468048630468049804304860469848902465048904165401984065044681435135406801685046468048601534086041530468045315487972021596172958763290756783095027617429617290489708016504860165878941336048134841;
Log[10,n] //N
p=NextPrime[n]
Out[2]= 300.219
Out[3]= 1654486103240894053403206870530135065304612049816540984530749803013049804150490864068468743049435046830646806468048630468049804304860469848902465048904165401984065044681435135406801685046468048601534086041530468045315487972021596172958763290756783095027617429617290489708016504860165878941336048134883

Then p is your pet prime, rinse and repeat ad nauseam.

Edited May 25, 2015 by jchd

BinaryBrother · May 25, 2015

Mathematica would be a sweet approach, but looks like it's a paid service.

That email was pretty interesting. ^.^

jchd · May 25, 2015

I'm a proud owner of a Mathematica licence, which is indeed a payware product, but you can still use Wolfram Alpha, for free, from desktop or mobile devices:

https://www.wolframalpha.com/input/?i=NextPrime[1647816546168546816416846816898646813546864135068068406801680468578160460436]

BinaryBrother · May 26, 2015

Someone needs a Math UDF based on Wolfram! lol

However, upon testing, it looks like the input is fairly limited. Wolfram refuses to compute an input of 200 digits or more. [Could be less, I only started with a 200 digit prime.]

I'm thinking about generating all primes between 300-500 digits, "packing" them into an AutoIt variable and working from that. I would say that's not the most effeciant method, but with AutoIt... It just might be. ^.^

jchd · May 26, 2015

First I'm surprise when you say that Wofram Alpha barks at large numbers: to wit https://www.wolframalpha.com/input/?i=NextPrime[10^300]

Now, please carefully reread the newsgroup answer I cited above. Digest it and then realize that you'll going to need a fabulous number of Universes just to store each 512-bit (155-digit) prime in a single hydrogen atom!

A circa 300-digit prime is a circa 995-bit number.

A simple calculation gives:

In[1]:= n=10.0^300;
appPi=n/(Log[n]-1)

Out[1]= 1.44975*10^297

In[2]:= n=10.0^500;
appPi=n/(Log[n]-1)

Out[2]= 8.69344066468389*10^496

Given that our whole Universe contain approximately 1E90 hydrogen atoms, you're going to need roughly 1.45 10^207 Universes just to store all 300-digit primes.
Then a little more to store 301-digit primes, then even more to store 302-digit primes, and so on.

The real question is: how many Universes is AutoIt capable to manage?

Do you now realize?

Edited May 26, 2015 by jchd

BinaryBrother · May 26, 2015

On Wolfram, I forgot to use exponents and just threw a 300 digit integer at it! My fault. ^.^

Ahhh, I didn't understand that until now. So, too many 300 digit primes exist to 'cache' them in any possible way...

I am having such a hard time finding a local method to simply generate ~300 digit random primes, within a reasonable amount of time, for the DH key-exchange! Who would have thought that after I was virtually done with the implementation in 2 languages, they would be like, "Oh... by the way... The input primes need to be more than 300 digits. All of the examples using 2 digit primes, in every example in 10 different languages, are irrelevant." lol

jchd · May 26, 2015

Welcome to the wonderful world of crypto-size arithmetic and number theory!

Kid sister secrecy and strong crypto are indeed different worlds, needing different tools and once in strong crypto realm, you lose common sense due to the incredible size of numbers you need to manipulate, even it seems so easy to talk about 500-digit (or more) numbers.

Unfortunately with have at our disposal notations which hide the groundplay we are on: writing 10^50000 is easily understandable (even AutoIt deals with it with BigNum) but actually getting some grasp of what we're talking about is a completely different beast. Say, how many primes betwwen 10^50000 and 10.1^50000 seems a reasonable question at first. At first only, because you just can't give an exact answer to it and the best approximation you come up with is itself something ous brain can't fully realize!

Whatever common sense tells you about small samples often becomes a solid concrete block once in the vicinity of 1024-bit values, or beyond. Things as simple as enumerating integers seem overly simple for values you can count instictively (up to 5 in general), still you can manage more with a computer and typical For loops, but with crypto-size numbers you lose control of the consequences of what intuition had told you over years.

That was the whole point of the funny quote and the reason why it went thru to us today. Not only the number of hydrogen atoms in our own universe is way beyond common sense, but it itsself reveals sooo ridiculously negligible in face of crypto-size quantities and number-theoretic properties.

All of this tells you something: only theory built on solid grounds allows you to assert things about numbers when you let them reach really large values.

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_GetPrime() returns larger primes, 300 digits+

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