The prime from CP_Problem8_21 From: d.broadhurst@open.ac.uk Date: Tue Nov 6, 2001 7:05 pm Subject: Crandall-Pomerance Problem 8.21 solved by Primo Research Problem 8.21 on page 398 of the "Prime Numbers" book by Richard Crandall and Carl Pomerance asks for a proof of my claim that the following integral yields a prime number: P3139= 2^903*5^682/514269*int(x^906*sin(x*log(2))*f(x),x=0..infinity) f(x)=(1/cosh(Pi*x/5)+8*sinh(Pi*x/5)^2)/sinh(Pi*x/2) It is straightforward to evaluate the integral numerically, by expanding f(x) in powers of exp(-Pi*x/10) and integrating each term analytically. By taking 5000 terms, at 3500-digit precision, the following Pari program showed that P3139 differs from a 3139-digit integer by less than 10^(-100): \p3500 nl=5000;P=Pi/10;L=log(2); s=[1,1,0,-2,1,1,0,0,-1,1,0,0,1,-1,0,0,1,1,-2,0]; is(n)=4*n!*sum(k=1,nl,s[k%20+1]*imag(1/((2*k-1)*P-I*L)^(n+1))); approx_ans=2^903*5^682/514269*is(906) However that is not a proof that P3139 is an integer. To prove that it is, I used contour integration to express it as the real part of a rational polynomial of degree 1814 in z=(1+2*I)/5: p=12*19;n=4*p-5;z=(1+2*I)/5;a=x;for(k=2,n,a=x*(1-x)*deriv(a)) ans=2000^p/n/10!*real(2*(5/2)^n*subst(a,x,z^2)-4*subst(a,x,z)) Then Primo 1.1.0 proved primality. It took 178.5 hours to compute a path: Decimal size = 3139 Binary size = 10426 [....] Started 10/28/2001 05:59:47 PM Running time 178h 38mn 1s Backtracked test(s) = 3 Prime path computed and a further 36.5 hours to produce the certificate: Started 11/05/2001 04:49:58 AM Running time 36h 23mn 11s Candidate certified prime Finally, Cert_Val validated the certificate: Prime number being certified was: N = 32558766036484382179057335167515043164743683267363\ 25223822795341514785971380063717474609230277880574\ 83954642423671545762856106558143470932892214339297\ 83262070006022200423595177903221906317838723523745\ 32144128419670759806062371523123596023540051770348\ 06848351181404016546960130137932798661205678543923\ 77535812954800456692462625041234973875996239709830\ 98209718533017000346668995062659761388590878757602\ 16266827103166385014330020031109506160320811564489\ 49784129664931700481391438001939356295700579114601\ 61861264485610857438272977199455846043682729454582\ 98414978884362679303341990299401634752124104200222\ 21511518588081625544109499512545423423296106616565\ 87787623689334982805440502746976014236035982993789\ 36502988818757192980656635879431724957556906173128\ 91392750442848556228640237205778480765249541769022\ 85833585050492320528187453533912711763217407002725\ 55021503684496937046208544750316154065742574995526\ 04112540692086550558702273072789295027392666889847\ 10775421779816557436416104122254653865475231280437\ 54187246059112757187215284068344648244858461845295\ 03725872003976712956176412524407850537736128255155\ 36878910519533957535004524039565343689244000397325\ 81275152648092645656412780194306026431389838473030\ 98668648140986212200769873098313495296306160187995\ 70595799962259624939334537561002242889472999383703\ 54105052209975909429487579096222450415143145539113\ 27614246733223407596062358665970292075620356051485\ 73123639235763983930033714801695665724374734690062\ 36753341533730423792137209231637973687522089194420\ 10590083371880083665144679593642231114528576292538\ 73088151883745442673180955428838740937042273360155\ 52190286399496506733899253301028178247762867371274\ 93316187556535764761736730099729853448253262051125\ 91352435544227565432598109496117323365875286744914\ 20661609857390422420038808245806198472990182916458\ 61287145133009280422927440437304943088220998086663\ 23078663065279519707594057439986088836939654172407\ 94389689300809965774751255348997730537154556925621\ 02298281179322629970400544019265145518040630062608\ 81393714099639305310125181025080507516823314960739\ 92712279583980805651224403925464350141896599762320\ 63408402832825896798751329770493277601551910573715\ 99043603773523745902829325385560883460746910829021\ 62953970268343360127966662204538193434225680342857\ 07667053913030598148106732033376752031579602604463\ 97737736180590235090610234450201927927515960525334\ 63515066035637928621988922219461171842946334341804\ 94920986536404519768268322105048625103184552393112\ 69970522503039105564509333129030152812650813197791\ 68873322539174822214308954124728707084306559757254\ 63992615177189354526385546635918037708098366051127\ 98396106368537059768519075888257545103418263709588\ 33767839083150935164847385266837023578943388779080\ 98211560553669854138746214406322210600434114097234\ 08610918089003177287937331560133568823370159284594\ 95350678721197827993116611550571082465967137841794\ 84438342036666721675241675495512041516289435948981\ 33577355175375246629666444955461608537279282522191\ 31197117180743669676647000824033814233519472952050\ 07735517890092032953543595785022124784607248331744\ 73263978933377296484717074115827628656714851203300\ 795384284341450842087568434052915829849 Certificate for this number was FULLY validated! Total time used to validate certificate: 5h 17mn 2.200s There were 471 steps in the primality proof David Broadhurst