Problems & Puzzles: Puzzles

Puzzle 553.- SCN=PCP

JM Bergot sent the following puzzle.

One notices that the sum of the closed interval between the two consecutive primes 19 and 23, 19+20+21+22+23= 105=3*5*7, equals the product of consecutive primes.

Q.  Larger such intervals exist?

Contributions came from W. Edwin Clark. Torbjörn Alm, Fred Schalekamp, Farid Liam, Giovanni Resta & Jan van Delden.

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W. Edwin Clark. Torbjörn Alm, Fred Schalekamp, Farid Liam found one more solution:

229+230+231+232+233=3*5*7*11

***

Giovanni Resta wrote:

The next two such consecutive primes are 239 and 239+4=243,
where S = 239+240+241+242+243 = 3x5x7x11 (4 terms)
and (p,p+k) with p=10131543901 and k=12 where the sum S
is equal to 13 x ... x 41 (8 terms).

I found these other longer sequences, where
the prime p can be expressed as p=(2S - k^2 - k)/(2k+2).
The longest sequence has 391 consecutive prime
factors and in the pair of consecutive primes
(p,p+k) the prime p has 988 digits.

S =   3 x...x   11 (  4 terms) k=   4 (p=239)
S =  13 x...x   41 (  8 terms) k=  12 (p=10131543901)
S = 277 x...x  383 ( 18 terms) k= 312 (p has  43 digits)
S = 613 x...x  881 ( 41 terms) k= 876 (p has 115 digits)
S =  17 x...x  353 ( 65 terms) k= 240 (p has 137 digits)
S = 233 x...x  907 (105 terms) k= 768 (p has 283 digits)
S =1621 x...x 2671 (131 terms) k=2376 (p has 433 digits)
S = 653 x...x 1619 (138 terms) k= 732 (p has 417 digits)
S =  43 x...x  887 (141 terms) k= 180 (p has 357 digits)
S =  11 x...x  857 (144 terms) k=1512 (p has 351 digits)
S =1039 x...x 2137 (148 terms) k=1212 (p has 470 digits)
S = 787 x...x 1889 (153 terms) k=1452 (p has 473 digits)
S = 317 x...x 1439 (163 terms) k= 372 (p has 470 digits)
S =  31 x...x 1039 (165 terms) k= 228 (p has 425 digits)
S = 593 x...x 2129 (213 terms) k= 660 (p has 658 digits)
S = 349 x...x 2677 (319 terms) k=2520 (p has 988 digits)

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Jan van Delden wrote:

I only investigated the situation where  4|g and g+1=prime.

Call the two primes consecutive primes p1,p2=p1+g, with  gap g.

Gap g,[First term product, ..,Last term product], Number of primes in product, Smallest prime p1:

4,[3,..,7],3,19
4,[3,..,11],4,229
12 ,[13,..,41],8,10131542901
312,[227,..,383],18,6644969653495363048790133051929586253470187
240,[17,..,353],65,
1275708898179173567335109688602441097614654634271401575583828007310160802264951615
0067088238340365650883565019307972166422735625015351887
180,[43,..,887],141,
7112804436280023418022416210308249485205911395522033711355563767426317902667124613
2080502385268384716493289760105736606882055606340353846569324824909312502041809696
5468504274664780131488089673269122599617821802428950424627385016608123319315084852
7218146106231486893668358955195976017017192834746372251148932332203986289859073730
52558458431153664109395832021
228,[31,..,1039],165,
1555958068516809224340924396675933931391916938320464284862842419011783664967984246
7503861616449440191109005042393408474235964193180176081395250518461955293184030281
6107779102243760573904189777131881147642552986165246049901945515955504264285618233
5785236658925195988302533726221204279917103715545185726595668617209860196143632667
3011567099263172573754410072294658513098344947883043129059013544892290652959154491
945785891289409

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