Problems & Puzzles: Puzzles
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Problems & Puzzles: Puzzles
 From May 2 to 8, contributions came from Raid Zaman, Jeff Heleen, Michael Branicky, Paul Cleary, Emmanuel Vantieghem, Simon Cavegn *** Raid wrote: For 1266 Q1, the OEIS sequence is already at https://oeis.org/A058941 (largest known number 21217).For 1266 Q2, https://www.angio.net/pi/ verifies that 57255323573532353 occurs at position 74391718. *** Jeff wrote: Q1. I found one other larger solution for n up to 4300 digits:Here is the full list for pi: pi 2, 17 7, 1415929 35, 14159265358979323846264338327950291 59, 14159265358979323846264338327950288419716939937510582097497 84, 141592653589793238462643383279502884197169399375105820974944592307816406286208998631 451, 14159265358979323846264338327950288419716939937510582097494459230781640628620899862\ 80348253421170679821480865132823066470938446095505822317253594081284811174502841027\ 01938521105559644622948954930381964428810975665933446128475648233786783165271201909\ 14564856692346034861045432664821339360726024914127372458700660631558817488152092096\ 28292540917153643678925903600113305305488204665213841469519415116094330572703657595\ 919530921861173819326117931051185483 *** Michael wrote: Q1 is already documented as OEIS A058941: Numbers k such that 3 + (integer formed from first k digits after decimal point in Pi) is prime. The terms are 0, 2, 7, 35, 59, 84, 451, 10090, 21217 with 451 from Patrick De Geest, Nov 25 2001 10090 from Metin Sariyar, Jul 02 2020 21217 from Michael S. Branicky, May 02 2025... I also reported that the next term is > 10^5 on May 05 2025 Q2: In the first 1B digits of Pi, I found the following larger prime-digit primes:Prime-digit prime ( # digits), Index (from 0) 27255557773375577 (17 digits), 179159273 35322327552733733 (17 digits), 241611789 37535773532553353 (17 digits), 644785476 55237333573275257 (17 digits), 560366338 57255323573532353 (17 digits), 74391718 57537552355725773 (17 digits), 718103626 77257735355273353 (17 digits), 373203990 357537552355725773 (18 digits), 718103625 523237352372723557 (18 digits), 188438685 7232553322522755523 (19 digits), 896631794 For Q3, the next pi-semiprime term is3141592653589793238462643383279502884197 = 59 * 5324733311169141082140073530982208278 (non brilliant)ASIDE: There is a typo in the question: 314159265358 = 2 × 157079632679 Using factordb.com, I further found the following pi-semiprimes (all non-Brilliant): 104 digits:31415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821= 19595256920847214432055206117* 1603241369214956083288729535993167375753383068787843721300976529752490235913 112 digits:3141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513= 1215666422974078739455530964256912318288897* 2584255511395974781222544726762221278238062155282351848420786629580529 260 digits:3141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844 6095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669= 3561511* 8820954515063390899151072068230318210998560440709310798071224804044733839896069389166662198550326162076006746892771934853195299999 215451620745687457173320867042904269288901595714739178463716537532474178596699478580673894109763098397805827822155264743379 346 digits:314159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460 955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316527120190914564856 6923460348610454326648213393607260249141273724587006606315588174881520920962829254091715= 5* 628318530717958647692528676655900576839433879875021164194988918461563281257241799725606965068423413596429617302656461329418768921 910116446345071881625696223490056820540387704221111928924589790986076392885762195133186689225695129646757356633054240381829129713 384692069722090865329642678721452049828254744917401321263117634976304184192565850818343 402 digits:314159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460 955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316527120190914564856 692346034861045432664821339360726024914127372458700660631558817488152092096282925409171536436789259036001133053054882046652138414 695194151160943= 2630158375941188689* 1194449992945991480655892599545764793842838090522138145584921313184364107869382173095867823779260501604608836385587662405130677 40108656122082943833265469519892796023689194663811874026545417227133938562721978380602936217134387200948705399475130667373478999 692462463575137712442381232944292544874554110084531867146825077253701761255863918024332514565291833338840508268048027383578705087 568 digits:31415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446 09550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648 56692346034861045432664821339360726024914127372458700660631558817488152092096282925409171536436789259036001133053054882046652138 41469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624 40656643086021394946395224737190702179860943702770539217= 81136751403599* 3871972440654606921227418134200191931208259456849440194770890518668229618716703285029995265639742557368119808496059252797888913 38500713036805039348196327645431762477873996715786516962596974458123514525743035615035853521771278602824157883896847363732961864 96806556703558181363797396486166199036352213670126745912673174016616726013449427213962712670565660454690008692584914208628932303 3863964213586195997562160028289839881229293676666417509237479953748000986273904132992854060495278433695792197409179672676186244 03595449026783211508995693491818988470159583 *** Paul wrote: Q1. Aborted search at approx 24,000 In order:- 27355984 <-Example given.4511009021217 Q2. Checking up to 500,000,000 decimal digits.Prime found: length 17, position 74391718, value = 57255323573532353Prime found: length 18, position 188438685, value = 523237352372723557Q3. Checking primes up to length 5000.semiprime at length 260: 3141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982 1480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756 482337867831652712019091456485669 = 3561511 x 8820954515063390899151072068230318210998560440709310798071224804044733839896069 3891666621985503261620760067468927719348531952999992154516207456874571733208670429042692889015957147391784637165375324741785 96699478580673894109763098397805827822155264743379 factor digit lengths = {7,253} semiprime at length 346: 314159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798 214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847 5648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715 = 5 x 6283185307179586476925286766559005768394338798750211641949889184615632812572417997256069650684234135964296173026564613 2941876892191011644634507188162569622349005682054038770422111192892458979098607639288576219513318668922569512964675735663305 4240381829129713384692069722090865329642678721452049828254744917401321263117634976304184192565850818343 factor digit lengths = {1,345} semiprime at length 568: 3141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067 9821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461 2847564823378678316527120190914564856692346034861045432664821339360726024914127372458700660631558817488152092096282925409 1715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962 749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217 = 81136751403599 x 387197244065460692122741813420019193120825945684944019477089051866822961871670328502999526563974255736811980849605925279 78889133850071303680503934819632764543176247787399671578651696259697445812351452574303561503585352177127860282415788389684 73637329618649680655670355818136379739648616619903635221367012674591267317401661672601344942721396271267056566045469000869 25849142086289323033863964213586195997562160028289839881229293676666417509237479953748000986273904132992854060495278433695 79219740917967267618624403595449026783211508995693491818988470159583 factor digit lengths = {14,554} semiprime at length 691: 3141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067 982148086513282306647 093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786 78316527120190914564856692346034861045432664821339360726024914127372458700660631558817488152092096282925409171536436789259 03600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527 24891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132 000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279 = 23614939 x 1330341210531940496 8450875029910104295408806261896784154195547963548906081384368592389905497287615555484382519528347317124 5280142775324646475917613343336653016562274528930120086507936549103470262209102654417653827771202209265738447759140227517 4711189872499169666904643967301514794602430947316971321108949121997141858206017335687769916674040145399217151436465857485 9458796606760095892621814318222509951997154302900209024266766838217883741354543008214198963047840025119553990318385852663 9765539221250968712376236180015398010810927563539131934119783336296610535507652135674637701869667530151929152533704702797 162725902597539116479248274060314497261126896049506666149971991464663515979061 factor digit lengths = {8,684} Q4. Did not find another Brilliant Semiprime. *** Emmanuel wrote: Q1. Here are two more such (probable) primes : 141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982 148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428 81097566593344612847564823378678316527120190914564856692346034861045432664821339360726024914127372458 70066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572 703657595919530921861173819326117931051185483 (451 digits; Verified prime by Dario Alpern's Alpertron) And 141592653589793238462643383279502...982936804352202770984297 (10090 digits; Verified prime by Dario Alpern's Alpertron) I stopped the search at this point because of the almost unfeasible effort (for me) to get a proof of the primality (which, as far as I know, should be done by PRIMO). Q2. Here are some larger primes of that form : 27255557773375577 at position 179159273, 35322327552733733 at position 241611789 57255323573532353 at position 74391718, 523237352372723557 at position 188438685. To get larger primes we need more than 320000000 digits of \[Pi]. Q3. The next term is : 3141592653589793238462643383279502884197 (39 digits) = 59*53247333111691410821400735309822082783. *** Simon wrote: Q1. 3+ 451 digits 3+ 10090 digits 3+ 21217 digits Searched up to 156000 digits. Q2. 57255323573532353 starts at position: 74391718 523237352372723557 starts at position: 188438685 7232553322522755523 starts at position: 896631794 7737322357355773277 starts at position: 1328110884 272335752537275235533 starts at position: 2224940580 2757222273335222377723 starts at position: 10757602468 5723723273575332773557 starts at position: 65833408632 357575337577735335275353 starts at position: 94635087848 Searched only first 100B digits. Q3. 314 = 2 * 157 314159265358 = 2 * 157079632679 314159265358979323 = 317213509 * 990371647 Brilliant 3141592653589793238462643383279502 = 2 * 1570796326794896619231321691639751 3141592653589793238462643383279502884197 = 59 * 53247333111691410821400735309822082783 31415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821 = 19595256920847214432055206117 * 1603241369214956083288729535993167375753383068787843721300976529752490235913 3141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513 = 1215666422974078739455530964256912318288897 * 2584255511395974781222544726762221278238062155282351848420786629580529 314159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328 230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564 82337867831652712019091456485669 = 3561511 * 88209545150633908991510720682303182109985604407093107980712248040447338 39896069389166662198550326162076006746892771934853195299999215451620745687457173320867042904269288901595714739178 463716537532474178596699478580673894109763098397805827822155264743379 314159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328 230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097 566593344612847564823378678316527120190914564856692346034861045432664821339360726024914127372458700660631558817488 1520920962829254091715 = 5 * 628318530717958647692528676655900576839433879875021164194988918461563281257241799725606 965068423413596429617302656461329418768921910116446345071881625696223490056820540387704221111928924589790986076392 88576219513318668922569512964675735663305424038182912971338469206972209086532964267872145204982825474491740132126 3117634976304184192565850818343 31415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132 82306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475 64823378678316527120190914564856692346034861045432664821339360726024914127372458700660631558817488152092096282925 409171536436789259036001133053054882046652138414695194151160943 = 2630158375941188689 * 1194449992945991480655892599 545764793842838090522138145584921313184364107869382173095867823779260501604608836385587662405130677401086561220829 438332654695198927960236891946638118740265454172271339385627219783806029362171343872009487053994751306673734789996 924624635751377124423812329442925448745541100845318671468250772537017612558639180243325145652918333388405082680480 27383578705087 31415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132 82306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475 64823378678316527120190914564856692346034861045432664821339360726024914127372458700660631558817488152092096282925 40917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480 74462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539 217 = 81136751403599 * 38719724406546069212274181342001919312082594568494401947708905186682296187167032850299952656 397425573681198084960 59252797888913385007130368050393481963276454317624778739967157865169625969744581235145257430356150358535217712786 02824157883896847363732961864968065567035581813637973964861661990363522136701267459126731740166167260134494272139 62712670565660454690008692584914208628932303386396421358619599756216002828983988122929367666641750923747995374800 098627390413299285406049527843369579219740917967267618624403595449026783211508995693491818988470159583 ***
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