Problems & Puzzles: Puzzles

 Puzzle 468. Primes & Semiprimes Gaps JM Bergot sent one more nice puzzle: Q1. Between consecutive primes 113 and 127 thre are six  semiprimes: 115 (5*23), 118 (2*59), 119 (7*17), 121 (11*11), 122 (2*61), and 123 (3*41).  Can you find a prime gap with more than six semiprimes?  Q2. Between semiprimes 1502 (2*751) and the next one 1507 (11*137) there are no primes.  Can you find a larger gap in semiprimes with not one prime within?

Luis Rodrígues wrote:

I think, the larger the gap the bigger the quantity of semiprimes in it.
For example: The 112 gap between 370261 and 370373 contains
17 semiprimes. They are:
370000+267, 283, 289, 291, 297, 303, 307, 309, 313, 317, 321,
327, 339, 343, 347, 361, 369.
How many will contain the 1.000.000 gap found by Kruse Anderson?

***

J. K. Andersen wrote:

Q1.
Semiprimes are asymptotically denser than primes so the average number of
semiprimes between two consecutive primes will tend to infinite.
Exhaustive search shows that many of the maximal prime gaps at
http://hjem.get2net.dk/jka/math/primegaps/maximal.htm contain more
semiprimes than any earlier prime gap. For example the maximal gap of 464
starting at 42652618343 with 90 semiprimes.

The maximal gap of 1356 at 401429925999153707 contains 165 semiprimes.
This is the largest for the known maximal gaps, but it's possible there
are earlier non-maximal gaps beating that. The later non-maximal gap of
1358 at 523255220614645319 also contains 165 semiprimes. These two prime
gaps have the 3rd and 4th largest known merit at
http://hjem.get2net.dk/jka/math/primegaps/gaps20.htm#top20merit.
There are probably a lot more than 165 semiprimes in many known prime gaps
with lower merit between much larger primes, but counting the semiprimes
quickly becomes hard. There is no feasible known semiprime test for a large
number without a known prime factor.

Q2.
Two of the gaps which are larger than any gap between smaller semiprimes are
from 843 to 849, and of length 78 from 19423306039 to 19423306117.
It is easier to find large gaps between very large numbers.

Let n be the 2000-digit number
14927914859738396468339148731572317688893626075558750117261389272965251263751443\
38163512838785762692232151736515796008036966271110574257563685106687520985305071\
28620075667964146224410813241373409665299889683682850172954254249242767202932736\
35597220807174187324406526297413602921495911366209528544582779528467295565290113\
36971619027930672280449448520334002783088528255817702716629071559539825028751407\
79821298673224691065948391467141894849155662927311830153908740034019634196629176\
64090833098120284040803953451831849806419324080336252502761960068136335152712169\
64198044221580774430977487093695236490741188616926429573471932971230044376873284\
45288433168151708621387821171539986436684684559831220534445675302094711969804198\
30787997594712878492989625779425953054144898854091318496070569172699558351034072\
90514732387413240966864201042319434923532812314899737632959091420172363408088116\
37221461328505573462607571541766218480173652155083708133431027141471101211280139\
38671210907193818559693621501243250073165759097238048042893711707742757151153131\
92595060993048867029521578948650318377601811828665155557300268123603312884766003\
07199768481848331959192572367375332367343305354395681665115939094698806216351691\
81047300839864262050231687611596789896096265345418969320703070770303756654180714\
17684939939309341042712641999225152226939093076656340757730766016416170650397781\
76555123264157177889042521452988249255541470500086882704871293570836113237331473\
05114174728655116152909807136744355491420394679148178559407740108222920215222560\
74441399728974689265093451370643347379752895826237320806939491837373033525540818\
50003595155896917801120091030309119206091238558801578034508129101687729236827308\
51785333171479754859495672377560954714271949707636888165354156091413896432877852\
54649435295495401672093384571655465160919305386729582538255012971610819651240683\
81582965301490873327694537972816214770351608351273800433623299301755847896642433\
62309343049581064367597791160909853621253046519344218659768068065152038559943405

n = 5*p and n+13517 = 2*q are consecutive semiprimes. p and q are 1999-digit
primes proved by Marcel Martin's Primo. PrimeForm/GW performed prp tests.
It is relatively easy to show there are no primes or semiprimes between
n and n+13517, because the interval is constructed such that all numbers have
at least one prime factor below 7500. Around 2000 intervals were tested and
this was the largest found semiprime gap. In cases where a number only has
one small factor, a composite prp test of the cofactor will prove the number
cannot be a semiprime.

***

Farideh Firoozbakht wrote:

Answer to Q1: Between consecutive primes 11981443 and 11981587 there
are 40 semiprimes (11981446,11981449,11981453,11981458,11981461,
11981462,11981467,11981471,11981473,11981474,11981477,
11981479,11981483,11981491,11981497,11981499,11981501,
11981503,11981507,11981509,11981513,11981517,11981518,
11981521,11981527,11981531,11981533,11981539,11981542,
11981545,11981549,11981555,11981557,11981558,11981561,
11981569,11981573,11981577,11981579 & 11981581).

If we define a(n) as the smallest prime p such that between p and q , q is next-prime(p),
there are exactly n semiprimes then (n,a(n)) for n = 0, 1, 2, ..., 40 are:

(0,2),(1,3),(2,7),(3,31),(4,89),(5,139),(6,113),(7,211),(8,1381),(9,1637),(10,1129),(11,2557),(12,2971),(13,1327),(14,15683),(15,16141),(16,9973),(17,35677},(18,34061),(19,43331),(20,19609),(21,107377),(22,162143),(23,44293),(24,404597),(25,461717),(26,838249),(27,155921),(28,535399),(29,492113),(30,396733),(31,2181737),(32,370261),(33,1468277),(34,6034247),(35,3933599),(36,1671781),(37,25180171),(38,1357201),(39,3826019) & (40,11981443).

Answer to Q2: Between semiprimes 1000015323893 (11621*86052433) and the
next one 1000015323953 (131*7633704763) there are no primes and the gap is 60.

If we define b(n) as the smallest semiprime m such that the gap between m and the
next-semiprime(m) is n with not even one prime within, then (n, b(n)) for n = 1, 2, ..., 21
are:

(1, 9), (2, 49), (3, 62), (4, 403), (5, 341), (6, 843), (7, 6722), (8, 3473), (9, 2869),
(10, 14059), (11, 18467), (12, 26603), (13, 166126), (14, 41779), (15, 74491),
(16, 192061), (17, 463161), (18, 226489), (19, 344119), (20, 517421) & (21, 943606).

***

Carlos Rivera.

Sorry, but I found more amusing to seek for gaps of X-type numbers free of Y-type numbers:

a) There is a gap of 55 numbers between the primes 359589563 & 359589619 free of semiprimes.

b) There is a gap of 59 numbers between the semiprimes 2,272,713,503 & 2,272,713,563 free of primes.

J. K. Andersen wrote:

Regarding the new point a)
After the gap of 55 between 359589563 & 359589619, the next two gaps which are larger than any gap between smaller primes are: A gap of 57 between 5527105153 & 5527105211. A gap of 65 between 7940336347 & 7940336413.

It is easier to find large gaps for much larger numbers. There is a gap of 1005 between the 217-digit primes 33888513103*499#-503 & 33888513103*499#+503. Each number in the gap has at least one prime factor below 510.

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