Curios from the PPP pages for the G.L. Honaker & C. Caldwell collection Puzzle 2 3576 86312 64621 65676 29137 the largest left-truncatable prime Puzzle 3 1480028171 the central prime in the smallest prime-magic square using only consecutive primes (obtained first by H. Neslon) Puzzle 5: 61 is the largest prime known such that divides 67*71+1 (61, 67 & 71 are consecutive primes) By variation made by CR: 2 divides 3*5-1 11 divides 13*17-1 83 divides 89*97-1 Puzzle 7: 31513 is the first palprime obtained adding consecutive primes, the central one being a palprime too. 10499 + 10501 + 10513 = 31513 By extension by CR 41+43+47 = 131 first time that three consecutive primes add up to a palprime Puzzle 8 Primes concatenating natural numbers a) In reverse order 82 82818079787776757473727170696867666564636261605958575655545352515049484 74645444342414039383736353433323130292827262524232221201918171615141312111098765 4321 primes concatenating the first odd numbers a) in ascending order (notice that all ending odd numbers are primes. Is this necesary?) 13 135791113151719 135791113151719212325272931 135791113151719212325272931333537394143454749515355575961636567 1357911131517192123252729313335373941434547495153555759616365676971737577798183 85878991939597 b) in descending order 3 31 73 737169676563615957555351494745434139373533312927252321191715131197531 123 12312111911711511311110910710510310199979593918987858381797775737169676 563615957555351494745434139373533312927252321191715131197531 817 81781581381180980780580380179979779579379178978778578378177977777577377 17697677657637617597577557537517497477457437417397377357337317297277257237217197 17715713711709707705703701699697695693691689687685683681679677675673671669667665 66366165965765565365164964764564364163963763563363162962762562362161961761561361 16096076056036015995975955935915895875855835815795775755735715695675655635615595 57555553551549547545543541539537535533531529527525523521519517515513511509507505 50350149949749549349148948748548348147947747547347146946746546346145945745545345 14494474454434414394374354334314294274254234214194174154134114094074054034013993 97395393391389387385383381379377375373371369367365363361359357355353351349347345 34334133933733533333132932732532332131931731531331130930730530330129929729529329 12892872852832812792772752732712692672652632612592572552532512492472452432412392 37235233231229227225223221219217215213211209207205203201199197195193191189187185 18318117917717517317116916716516316115915715515315114914714514314113913713513313 11291271251231211191171151131111091071051031019997959391898785838179777573716967 6563615957555351494745434139373533312927252321191715131197531 Primes concatenating odd prime numbers in ascending order 3 23 7 2357 719 23571113171923293137414347535961677173798389971011031071091131271311371 39149151157163167173179181191193197199211223227229233239241251257263269271277281 28329330731131331733133734734935335936737337938338939740140941942143143343944344 94574614634674794874914995035095215235415475575635695715775875935996016076136176 19631641643647653659661673677683691701709719 1033 23571113171923293137414347535961677173798389971011031071091131271311371 39149151157163167173179181191193197199211223227229233239241251257263269271277281 28329330731131331733133734734935335936737337938338939740140941942143143343944344 94574614634674794874914995035095215235415475575635695715775875935996016076136176 19631641643647653659661673677683691701709719727733739743751757761769773787797809 81182182382782983985385785986387788188388790791191992993794194795396797197798399 1997100910131019102110311033 2297 23571113171923293137414347535961677173798389971011031071091131271311371 39149151157163167173179181191193197199211223227229233239241251257263269271277281 28329330731131331733133734734935335936737337938338939740140941942143143343944344 94574614634674794874914995035095215235415475575635695715775875935996016076136176 19631641643647653659661673677683691701709719727733739743751757761769773787797809 81182182382782983985385785986387788188388790791191992993794194795396797197798399 19971009101310191021103110331039104910511061106310691087109110931097110311091117 11231129115111531163117111811187119312011213121712231229123112371249125912771279 12831289129112971301130313071319132113271361136713731381139914091423142714291433 14391447145114531459147114811483148714891493149915111523153115431549155315591567 15711579158315971601160716091613161916211627163716571663166716691693169716991709 17211723173317411747175317591777178317871789180118111823183118471861186718711873 18771879188919011907191319311933194919511973197919871993199719992003201120172027 20292039205320632069208120832087208920992111211321292131213721412143215321612179 220322072213222122372239224322512267226922732281228722932297 Yves Gallot discovered (11/6/98) that the number 235711.....11927 - a list of prime numbers from 2 to 11927 - is prime (5719 digits long). Here is his communication Primes concatenating odd prime numbers in descending order 5 53 17 171311753 89 8983797371676159534743413731292319171311753 383 38337937336735935334934733733131731331130729328328127727126926325725124 12392332292272232111991971931911811791731671631571511491391371311271131091071031 01978983797371676159534743413731292319171311753 Puzzle 9 Giovanni resta found that: the sum of the primes from 2 to 3531577135439 (the 126789311423-th prime) is equal to 219704732167875184222756 which is the square of 468726713734. I double checked, so I hope this is correct. This is the largest known case (June 2003). The earliest is 2+3+5+7+11+13+17+19+23 =10^2 By extension by CR 2 consecutive primes add up to a cube:. 3+5=2^3 =8 107+109=6^3=216 3 consecutive primes add up to a cube 439 + 443 + 449 = 11^3 = 1331 4 consecutive primes add up to a fouth power 5171 + + + 5197 = 12^4 = 20736 5 consecutive primes add up to a fifth power 41 + 43 + 47 + 53 +59 = 3^5 = 243 61+67 =2^7 Puzzle 12 Jud McCranie has found that 902659997773 is the smallest prime whose reciprocal has period is 666. Puzzle 14: Paprime as sum of squares of consecutive numbers 1262^2 + 1263^2 = 3187813 (found independently by McCarnie and Patric De Geest) This is the largest known (Jun 2003 Check this with patrick) Puzzle 16: 452942827 is the first of 11 consecutive prime numbers all ending in "7" Puzzle 17: The smallest weekly primes (any prime that lost his primality condition by changing - one at a time-anyone of it’s digits to any number (0-9) other than the current. ) is 294001(Obtained by Ken Duisenberg) Puzzle 19 Patrick De Geest (28/12/98) has found that in a digital clock we can see 211 primes. Here are all of them Puzzle 21 7 is the smallest happy prime (7 49 97 130 10 1) 28999999999 is the smalles primes such that produces 6 primes in a row applying the happy procedure: 28999999999, 797, 179, 131, 11, 2 Puzzle 22 277777788888989 is the smallest prime with peristence 11 (Jud McCranie) 50006393431 Puzle 24 50006393431 is the earliest prime that remains prime in bases 2->9. Found by kack Brennen In base 10: 50006393431: In base 9: 153060758677 In base 8: 564447201127 In base 7: 3420130221331 In base 6: 34550030320411 In base 5: 1304403114042211 In base 4: 232210213100021113 In base 3: 11210002000211222202121 In base 2: 101110100100100111010000001001010111 Giovanni resta found 3 mores cases: 727533146383, 2250332130313 & 2651541199513 Puzzle 26 30103 = average divisors of (149645) = average divisors of (179574), DE Geest & McCranie Puzzle 35 1999 is the least prime number such that the sum of its digits is a perfect number (28) 29999999999999999999999899999999999999999999999999999999 is the least prime such that SOD=496 THIS IS THE EARLIEST POD(23)=6 This is the largest POD(47)=28 Puzzle 36 19.972.667.609 starts a sequence of 7 self-describing primes Puzzle 41 347182965 /(3+4+7+1+8+2+9+6+5)= 7715177 (prime!) A9D\the Beast =Palindrome has only two solutions. One of these is prime: 913572846\666 = 1371731(Palprime!) Puzzle 46: 48205429 is expressible a a sum of consecutive primes in 9 ways (J. C. Meyrignac). No solution is known for 10 ways. Puzzle 47 The smallest prime such that 353^4 = 30^4 + 120^4 + 272^4 + 315^4 Puzzle 48 The smallest prime such that 709^3 = 193^3 + 461^3 + 631^3 Puzzle 58: 1627, 1637, 1657, 1663 & 1667 are 5 consecutive prime numbers all such that 4p^2+1 is also prime. No secuence of six consecutive primes is known.