Problems & Puzzles: Puzzles

Puzzle 1111 quadruplets of consecutive primes JD Bergot sent the following nice puzzle: Call these four consecutive primes p,q,r,s being these four primes in the same decade, and find (2*p+q)/5 and (r+2*s)/5 so that both result in a prime after division  by 5. Examples: (p,q,r,s)=(11,13,17,19) (p,q,r,s)=(55331, 55333, 55337, 55339) Q. Other quadruplets like these ones?

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During the week 12-18 Nov. 2022, contributions came from Paul Cleary, Michael Branicky, Alain Rochelli, Giorgos Kalogeropoulos, Gennady Gusev, Emmanuel Vantieghem, J-M Rebert, Ken Wilke

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Paul wrote:

Here are solutions with primes <=2280176348

{11, 13, 17, 19}

{55331, 55333, 55337, 55339}

{442571, 442573, 442577, 442579}

{845981, 845983, 845987, 845989}

{1065011, 1065013, 1065017, 1065019}

{1137881, 1137883, 1137887, 1137889}

...

9909986621, 9909986623, 9909986627, 9909986629}

{9915791861, 9915791863, 9915791867, 9915791869}

{9944201711, 9944201713, 9944201717, 9944201719}

{9983608631, 9983608633, 9983608637, 9983608639}

{9992335811, 9992335813, 9992335817, 9992335819}

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Michael wrote:

Such quadruplets are quite common. For example, there are already 2 more < 10^6 and 63 more such < 10^7:

(442571, 442573, 442577, 442579), (845981, 845983, 845987, 845989)

(1065011, 1065013, 1065017, 1065019), (1137881, 1137883, 1137887, 1137889), (2020721, 2020723, 2020727, 2020729), (2478521, 2478523, 2478527, 2478529), (3207431, 3207433, 3207437, 3207439), (3849821, 3849823, 3849827, 3849829), (5274671, 5274673, 5274677, 5274679), (6287921, 6287923, 6287927, 6287929), (7659011, 7659013, 7659017, 7659019), (9733811, 9733813, 9733817, 9733819), (9973211, 9973213, 9973217, 9973219), (12039821, 12039823, 12039827, 12039829), (12279011, 12279013, 12279017, 12279019), (12672131, 12672133, 12672137, 12672139), (14059811, 14059813, 14059817, 14059819), (14802581, 14802583, 14802587, 14802589), (14883221, 14883223, 14883227, 14883229), (14918081, 14918083, 14918087, 14918089), (14937821, 14937823, 14937827, 14937829), (15520781, 15520783, 15520787, 15520789), (15526031, 15526033, 15526037, 15526039), (16130831, 16130833, 16130837, 16130839), (17400281, 17400283, 17400287, 17400289), (17443961, 17443963, 17443967, 17443969), (19068521, 19068523, 19068527, 19068529), (19600871, 19600873, 19600877, 19600879), (21353531, 21353533, 21353537, 21353539), (25355711, 25355713, 25355717, 25355719), (27048311, 27048313, 27048317, 27048319), (27661721, 27661723, 27661727, 27661729), (29673731, 29673733, 29673737, 29673739), (30617261, 30617263, 30617267, 30617269), (31460831, 31460833, 31460837, 31460839), (31706531, 31706533, 31706537, 31706539), (34201961, 34201963, 34201967, 34201969), (34778831, 34778833, 34778837, 34778839), (39425921, 39425923, 39425927, 39425929), (40971521, 40971523, 40971527, 40971529), (41487071, 41487073, 41487077, 41487079), (42208211, 42208213, 42208217, 42208219), (44676761, 44676763, 44676767, 44676769), (45735161, 45735163, 45735167, 45735169), (54653231, 54653233, 54653237, 54653239), (54851471, 54851473, 54851477, 54851479), (55412171, 55412173, 55412177, 55412179), (56925221, 56925223, 56925227, 56925229), (67423121, 67423123, 67423127, 67423129), (69629171, 69629173, 69629177, 69629179), (70054211, 70054213, 70054217, 70054219), (70351571, 70351573, 70351577, 70351579), (71752271, 71752273, 71752277, 71752279), (72610121, 72610123, 72610127, 72610129), (74536661, 74536663, 74536667, 74536669), (75660581, 75660583, 75660587, 75660589), (77424161, 77424163, 77424167, 77424169), (77625131, 77625133, 77625137, 77625139), (83442761, 83442763, 83442767, 83442769), (83554061, 83554063, 83554067, 83554069), (88565921, 88565923, 88565927, 88565929), (91065971, 91065973, 91065977, 91065979), (91722011, 91722013, 91722017, 91722019), (96054311, 96054313, 96054317, 96054319), (99044711, 99044713, 99044717, 99044719)

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Alain wrote:

It is easy to show that the solutions belong to the sequence OEIS A007530.

By computer calculation I obtained the following values:

442571, 845981, 1065011, 1137881, 2020721, 2478521, 3207431, 3849821, 5274671, 6287921, 7659011, 9733811, 9973211, ...

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Giorgos wrote:

There are 44210 such quadruplets < 10^12.

I am sending all the starting primes in a txt file.

Here are some bigger quadruplets

{100000000420038281,100000000420038283,100000000420038287,100000000420038289},

{1000000000081854971,1000000000081854973,1000000000081854977,1000000000081854979},

{10000000000390435661,10000000000390435663,10000000000390435667,10000000000390435669},

{100000000000699512731,100000000000699512733,100000000000699512737,100000000000699512739}

{10000000000000646125661,10000000000000646125663,10000000000000646125667,
10000000000000646125669},

{10000000000000000843038161,10000000000000000843038163,10000000000000000843038167,
10000000000000000843038169},

{1000000000000000000900654661,1000000000000000000900654663,1000000000000000000900654667,
1000000000000000000900654669},

{1000000000000000000000000000017840310961,1000000000000000000000000000017840310963,
1000000000000000000000000000017840310967,1000000000000000000000000000017840310969}

 

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Gennady wrote:

461 prime quadruplets were found among the first 100000000 primes (see attached file Pu1111GG.txt)

And the smallest 30 digits prime quadruplet with the same properties are:

10^29+3184791181,+2,+6,+8  and the sums 6*10^28+1910874709,+4.

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Emmanuel wrote:

There are  281  quadruplets that satisfy the conditions of the puzzle.
Here are the first elements of them :
11, 55331, 442571, 845981, 1065011, 1137881, 2020721, 2478521, 3207431, 3849821, 5274671, 6287921, 7659011, 9733811, 9973211, 12039821, 12279011, 12672131, 14059811, 14802581, 14883221, 14918081, 14937821, 15520781, 15526031, 16130831, 17400281, 17443961, 19068521, 19600871, 21353531, 25355711, 27048311, 27661721, 29673731, 30617261, 31460831, 31706531, 34201961, 34778831, 39425921, 40971521, 41487071, 42208211, 44676761, 45735161, 54653231, 54851471, 55412171, 56925221, 67423121, 69629171, 70054211, 70351571, 71752271, 72610121, 74536661, 75660581, 77424161, 77625131, 83442761, 83554061, 88565921, 91065971, 91722011, 96054311, 99044711, 104736761, 105555971, 106811981, 108942011, 111356381, 114412931, 116129471, 118508771, 118783031, 118946621, 119732021, 121267961, 124482221, 129934661, 129977711, 130185821, 131409911, 139045721, 146561411, 146614331, 149875631, 151220681, 152192561, 153379271, 155761511, 161961971, 162877571, 165601481, 172139831, 173568671, 179515871, 180657221, 183146981, 183749471, 187408511, 188411471, 197913761, 199414211, 201042131, 201635381, 216099131, 221238461, 227832881, 229723721, 231198761, 233420771, 236893331, 237314171, 240253961, 241958531, 243005171, 248806421, 257874431, 260437271, 272995271, 276091721, 283685531, 293420711, 299742761, 304512911, 313419011, 318405881, 321162971, 324283361, 332497931, 335803331, 337031831, 339700721, 340893731, 347001371, 350586281, 351116531, 355281671, 355692011, 359772311, 367179011, 369268721, 380370371, 381662921, 382995161, 390295811, 396695771, 398713031, 400920131, 413457761, 420851021, 425729531, 430030121, 431422631, 441193931, 446609411, 452001371, 465138761, 466882181, 470789021, 474190811, 485436311, 494381681, 494660981, 495203411, 496012331, 496920371, 507676571, 508113581, 508469321, 508654961, 509037161, 518480861, 521627921, 524792621, 529982771, 537603881, 538307171, 538315571, 541153511, 547535621, 560395811, 561343331, 563518721, 568705721, 569065031, 572364971, 575996711, 577710311, 578246231, 578759471, 579192071, 581872931, 584183981, 585018521, 590703011, 595730411, 597270131, 600943661, 623987381, 625465361, 628880381, 629706521, 630220181, 631684511, 634436981, 637736711, 638954081, 649564961, 657222611, 658830581, 662004521, 665497031, 673556831, 679241111, 680786081, 682241171, 703520261, 710068061, 713584511, 719546411, 732705431, 741227021, 742668461, 746000111, 750559211, 752772611, 762824261, 764363981, 768904181, 771235181, 771557531, 774951761, 778986911, 787985831, 790103261, 800546561, 801397271, 803734361, 812169221, 817217831, 820916771, 827094131, 844792511, 850739711, 854537771, 854814131, 861460211, 862563131, 871922411, 873901661, 876384071, 880657361, 883605131, 890802461, 890954081, 894661211, 896627861, 897879881, 899713181, 904825211, 906667961, 916487981, 928992431, 935450561, 936195431, 946858811, 954248921, 955316771, 958732631, 962180831, 973483871, 979359671, 979662911, 981051431, 983866061, 984906821, 991868321, 995999021
Since I presumed that there are many bigger solutions, I restricted myself to find the smallest solution of  n  digits.  This is what I found :
n        first of the quadruplet
10      1001503961
11      10003198061
12      100012510181
13      1000007025461
14      10000007905331
15      100000046109911
16      1000000001137931,
17      10000000253890181
18      100000000420038281
19      1000000000081854971
20      10000000000390435661
21      100000000000699512731
22      1000000000001388248411
23      10000000000000646125661
24      100000000000001013693161
25      1000000000000001328123071
26      10000000000000000843038161
27      100000000000000001261922761
28      1000000000000000000900654661
29      10000000000000000004812583411
30      100000000000000000003184791181
31      1000000000000000000001969622671
32      10000000000000000000016115776411
33      100000000000000000000002675312011
34      1000000000000000000000031059124681

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J-M wrote:

n number of quadruplets like these ones starting with p < 10^c in attached file bpuzzle1111.txt.

    

c n

1 0

2 1

4 2

5 9

6 62

7 429

8 2915

9 20119

10 143710

 

Quadruplets :

[11, 13, 17, 19]

[1151, 1153, 1163, 1171]

[33071, 33073, 33083, 33091]

[33637, 33641, 33647, 33679]

[55331, 55333, 55337, 55339]

[57637, 57641, 57649, 57653]

[75997, 76001, 76003, 76031]

[90821, 90823, 90833, 90841]

[97007, 97021, 97039, 97073]

See more in Pu1111JMR.txt

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Ken wrote:

This problem involves quadruplets of consecutive primes (p, q, r, s) which are equivalent to finding quadruplets (p, p+2, p+6, p+8) where each of p, p+2, p+6 and p+8 are consecutive primes. Then (2*p+q)/5 and (r+2*s)/5 become x=(3*p +2)/5 and y=(3*p+ 22)/5 respectively where x and y are integers. P can’t be 2 or 3, because p divides p+6. For p=5, we get the quadruple (5, 7, 11, 13) but here the x and y derived are not integers. If p =7, then p + 2 is not prime, hence we must have p >=11.

 Considering the reduced residues (mod 30) which are 1, 7, 11, 13, 17, 19, 23 and 29, we must have (p, p+2, p+6, p+8) = (11, 13, 17, 19) respectively (mod 30). Then let p = 30t + 11 for some nonnegative integer t. Considering possible residues for p (mod 7), we can exclude  p = 0,1,5 or 6 because when one solves the respective congruences of the form

p = 30 t +11 = 0, 1, 5 or 6 (mod 7) where t is some nonnegative integer t and combines the solutions and reduces them to residues (mod 210), not all entries of the quads of the form (p, p+2, p+6, p+8) are primes. Hence using solutions of the respective congruences of the form

p = 30 t +11 = 2, 3, or 4 (mod 7) where t is is some nonnegative integer t and combines the solutions and reduces them to residues (mod 210), we obtain the following three cases of decades:

(p, q, r, s) = (p, p+2, p+6, p+8) – (where each of p, p+2, p+6 and p+8 are consecutive primes  (mod 210 )

(p, q, r, s) = (191, 193, 197 199) (mod 210)

(p, q, r, s) = (101, 103, 107 109) (mod 210)

(p, q, r, s) = (11, 13, 17 19) (mod 210).

 

Case I. (p, q, r, s) = (p, p+2, p+6, p+8) = (11, 13, 17 19) (mod 210).

Here x = (2*p + (p+2))/5 = (35 + 210k)/5 for some nonnegative integer k . Hence x = 7 + 42k so that x is divisible by 7 for all nonnegative integers k > 0. The only solution from this case is (p, q, r, s) = (p, p+2, p+6, p+8) = (11, 13, 17 19) with x = 7 and y = 11.

Case II. (p, q, r, s) = (p, p+2, p+6, p+8) = (p, q, r, s) = (191, 193, 197 199) (mod 210).

Here y = (2*(p + 8) + (p+6))/5 = (595 + 210k)/5 for some nonnegative integer k). Hence

y = 119 + 42 k so that y is divisible by 7 for all nonnegative integers k > = 0 since          119 = 7* 17. Hence y cannot be prime and this case yields no solutions.

Case III. (p, q, r, s) = (p, p+2, p+6, p+8) = (101,103, 107,109) (mod 210).

Here x = 19 + 42k and y = 23 + 42k for some nonnegative integer k.

The following additional decade solutions were found using a sieve in UBASIC where the product P of p, p+2,p+6, p+8, x and y where x=(3*p +2)/5 and y=(3*p+ 22)/5 ,

was compared to a product Q of all primes less than a given limit and seeking those cases where gcd (P,Q) =1. These cases were tested for those in which all of components of P are primes. It is likely that there are an infinite number of solutions discoverable in this case. The solutions discovered are:

(p, q, r, s) = (55331, 55333, 55337, 55339) with x =33199 and y =33203

(104471, 104473, 104477, 104479) with x =62683 and y =62687

(442571, 442573, 442577, 442579) with x =265543 and y =265547

(845981, 845983, 845987, 845989) with x =507589 and y =507593

(1065011, 1065013, 1065017, 1065019) with x =639007 and y =639011

(1137881, 1137883, 1137887, 1137889) with x =682729 and y =682733

(2020721, 2020723, 2020727, 2020729) with x =1212433 and y =1212437.

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