Problems & Puzzles: Puzzles

 Puzzle 499. 6*7*8*9 - 1 is prime JM Bergot sent the following puzzle: 6*7*8*9 - 1 = 3023 (prime) 7*8*9*10 - 1 = 5039 (prime) 8*9*10*11 - 1 = 7919 (prime) [9*10*11*12 -1 = composite] Q. Can you find a sequence like this getting more than 3 primes in a row?

One comment first: While this puzzle was intended in order to multiply just 4 consecutive numbers and getting more than 3 primes in a row, puzzlers experimented with other the product of other quantity of consecutive numbers... which I accepted.

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Well... I wonder why this puzzle has been so popular, now and the first time it was published before.

Contributions came from J. K. Andersen, Robin Garcia, Farideh Firoozbakht, Qushunliang, Shyam Sunder Gupta, Arkadiusz Wesolowski, Seiji Tomita, Jeff Heleen, Michael Dressner & F. Schneider.

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Andersen remembers me that this puzzle was first published as Puzzle 191 as proposed by Jean Brette. That time (2002?) contributions came from Felice Russo, Daniel Gronau, Pavlos N, Igor Schein, Jens Kruse Andersen, Ken Wilke & Phil Carmody.

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Robin wrote:

Consider primes of the form n(n+1)....(n+t-1)-1 for t=2,3,4,... and 5 consecutive primes in a row like this:
(Search with Pari gp to n=10^7)

For t=2
[2,3,4,5,6] is the only 5 consecutive solutions because
2*3-1=5
3*4-1=11
4*5-1=19
5*6-1=29 are all prime. No more found to n=10^7.
In fact [2,3,4,5,6,7] is a solution (6 in a row)

For t=3 no solution to n=10^7

For t=4

[6985660, 6985661, 6985662, 6985663, 6985664] and no more to 10^7
because with n=6985660
n(n+1)(n+2)(n+3)-1 and
(n+1)(n+2)(n+3)(n+4)-1 and
(n+2)(n+3)(n+4)(n+5)-1 and
(n+3)(n+4)(n+5)(n+6)-1 and
(n+4)(n+5)(n+6)(n+7)-1 are prime.

For t=5

[7432630, 7432631, 7432632, 7432633, 7432634]

because
7432630*7432631*7432632*7432633*7432634-1 and
7432631*7432632*7432633*7432634*7432635-1 and
7432632*7432633*7432634*7432635*7432636-1 and
7432633*7432634*7432635*7432636*7432637-1 and
7432634*7432635*7432636*7432637*7432638-1 are prime.
No more to 10^7.

For t=6

16 solutions to n=10^7
[274319, 274320, 274321, 274322, 274323, 274324] (6 in a row)
[508520, 508521, 508522, 508523, 508524]
[539996, 539997, 539998, 539999, 540000]
[737612, 737613, 737614, 737615, 737616]
[1501454, 1501455, 1501456, 1501457, 1501458]
[1601875, 1601876, 1601877, 1601878, 1601879]
[1820953, 1820954, 1820955, 1820956, 1820957]
[2079431, 2079432, 2079433, 2079434, 2079435]
[2304260, 2304261, 2304262, 2304263, 2304264]
[4205818, 4205819, 4205820, 4205821, 4205822]
[4244518, 4244519, 4244520, 4244521, 4244522]
[4347556, 4347557, 4347558, 4347559, 4347560]
[6324012, 6324013, 6324014, 6324015, 6324016]
[6565550, 6565551, 6565552, 6565553, 6565554]
[6890763, 6890764, 6890765, 6890766, 6890767]
[9406210, 9406211, 9406212, 9406213, 9406214]

For t=7

[3337080, 3337081, 3337082, 3337083, 3337084]
[9179517, 9179518, 9179519, 9179520, 9179521]

For t=8

[4662863, 4662864, 4662865, 4662866, 4662867]

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Farideh wrote:

If f(n) = n(n+1)(n+2)(n+3) - 1 then for each n, 7 divides one of the seven numbers
f(n), f(n+1), ... & f(n+6). (*)

First proof of (*) :
For each n, {f(n), f(n+1), ..., f(n+6)} is a complete set of residues mod 7.
So 7 divides one of the members of this set.

Second proof of (*) :
We can easily show that f(n)* f(n+1)* ...*f(n+6) = n^28+3n^22+6n^16+4n^14+
3n^10+6n^8+n^4+4n^2 (mod 7). Now according to the Fermat theorem n^28+3n^22+6n^16+4n^14+3n^10+6n^28+n^4+4n^2 = 14(n^2+n^4) (mod 7)
So f(n)* f(n+1)* ...*f(n+6) = 0 (mod 7) hence 7 divides of the seven numbers
f(n), f(n+1, ... & f(n+6).

So according to (*) at least one of the numbers f(n), f(n+1, ... & f(n+6) is composite.
Hence there is no set of requested primes with more than six terms.

Let's a(k) be the smallest number n such that all the k numbers
f(n), f(n+1), ..., f(n+k-1) are primes. According to (*) for k > 6, a(k) doesn't exist.

a(k) for k = 1, 2, ..., 6 : 1, 3, 6, 7046, 6985660, 512176360

a(6) = 512176360 so all the six numbers f(512176360), f(512176360+1), ...,
f( 512176360+5) are primes.

f(512176360) = 512176360*512176361*512176362*512176363 -1
= 68814209027966440530691697397299759

f(512176361) = 512176361*512176362*512176363*512176364 -1
= 68814209565392330534411826383274023

f(512176362) = 512176362*512176363*512176364*512176365 -1
= 68814210102818223686027471014025159

f(512176363) = 512176363*512176364*512176365*512176366 -1
= 68814210640244119985538643581785879

f(512176364) = 512176364*512176365*512176366*512176367 -1
= 68814211177670019432945356378788919

f(512176365) = 512176365*512176366*512176367*512176368 -1
= 68814211715095922028247621697267039

***

Qus Hun Liang wrote:

k=4:
24274*24275*24276*24277-1
24275*24276*24277*24278-1
24276*24277*24278*24279-1
24277*24278*24279*24280-1

k=5:
6985660*6985661*6985662*6985663-1
6985661*6985662*6985663*6985664-1
6985662*6985663*6985664*6985665-1
6985663*6985664*6985665*6985666-1
6985664*6985665*6985666*6985667-1

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Shyam wrote:

Smallest Sequence with 4 primes in a row:

7046*7047*7048*7049 - 1 = 2466836219235023 (prime)
7047*7048*7049*7050 - 1 = 2468236637185199 (prime)
7048*7049*7050*7051 - 1 = 2469637651311599 (prime)
7049*7050*7051*7052 - 1 = 2471039261783399 (prime)

There are about 102 more such sequences below 10^7.

Smallest Sequence with 5 primes in a row:

22083548*22083549*22083550*22083551 - 1 = 237834863103018338591192064599 (prime)
22083549*22083550*22083551*22083552 - 1 = 237834906182121950568459230399 (prime)
22083550*22083551*22083552*22083553 - 1 = 237834949261231414744158628799 (prime)
22083551*22083552*22083553*22083554 - 1 = 237834992340346731118820265023 (prime)
22083552*22083553*22083554*22083555 - 1 = 237835035419467899692974144319 (prime)

Smallest Sequence with 6 primes in a row but instead of product of 4 consecutive numbers

take product of 6 consecutive numbers:

274319*274320*274321*274322*274323*274324 - 1 = 426146652274254100141728695713919 (prime)
274320*274321*274322*274323*274324*274325 - 1 = 426155973101151418681825628015999 (prime)
274321*274322*274323*274324*274325*274326 - 1 = 426165294097938407995445090518799 (prime)
274322*274323*274324*274325*274326*274327 - 1 = 426174615264617545321599386655599 (prime)
274323*274324*274325*274326*274327*274328 - 1 = 426183936601191307926392037614399 (prime)
274324*274325*274326*274327*274328*274329 - 1 = 426193258107662173103017979851199 (prime)

***

274319*274320*274321*274322*274323*274324 - 1 is prime!
274320*274321*274322*274323*274324*274325 - 1 is prime!
274321*274322*274323*274324*274325*274326 - 1 is prime!
274322*274323*274324*274325*274326*274327 - 1 is prime!
274323*274324*274325*274326*274327*274328 - 1 is prime!
274324*274325*274326*274327*274328*274329 - 1 is prime!

***

Seiji wrote:

About Puzzle 499,I found several n.

Search condition.
n < 1000000
n*(n+1)*(n+2)*(n+3)-1 is prime
(n+1)*(n+2)*(n+3)*(n+4)-1 is prime
(n+2)*(n+3)*(n+4)*(n+5)-1 is prime
(n+3)*(n+4)*(n+5)*(n+6)-1 is prime

7046*7047*7048*7049-1=2466836219235023
7047*7048*7049*7050-1=2468236637185199
7048*7049*7050*7051-1=2469637651311599
7049*7050*7051*7052-1=2471039261783399

24274*24275*24276*24277-1=347274370961410199
24275*24276*24277*24278-1=347331596696099399
24276*24277*24278*24279-1=347388829502970023
24277*24278*24279*24280-1=347446069382604719

115811*115812*115813*115814-1=179896099284110874023
115812*115813*115814*115815-1=179902312721497101959
115813*115814*115815*115816-1=179908526319836531279
115814*115815*115816*115817-1=179914740079131941519

116345*116346*116347*116348-1=183237050485517979719
116346*116347*116348*116349-1=183243350268077970023
116347*116348*116349*116350-1=183249650213078849399
116348*116349*116350*116351-1=183255950320523410199

126852*126853*126854*126855-1=258946363912749032519
126853*126854*126855*126856-1=258954529219213660559
126854*126855*126856*126857-1=258962694718783058639
126855*126856*126857*126858-1=258970860411460271279

154040*154041*154042*154043-1=563055176877994605839
154041*154042*154043*154044-1=563069797890118158023
154042*154043*154044*154045-1=563084419186990811879
154043*154044*154045*154046-1=563099040768616264439

161066*161067*161068*161069-1=673025735618561598023
161067*161068*161069*161070-1=673042449903031779479
161068*161069*161070*161071-1=673059164498818701239
161069*161070*161071*161072-1=673075879405926228959

301136*301137*301138*301139-1=8223550477004515149023
301137*301138*301139*301140-1=8223659710712567384759
301138*301139*301140*301141-1=8223768945508832374679
301139*301140*301141*301142-1=8223878181393317346119

340568*340569*340570*340571-1=13453119718476886380239
340569*340570*340571*340572-1=13453277726507217789959
340570*340571*340572*340573-1=13453435735929408385319
340571*340572*340573*340574-1=13453593746743466340023

378775*378776*378777*378778-1=20584110609399548576399
378776*378777*378778*378779-1=20584327984998354195023
378777*378778*378779*378780-1=20584545362318828547719
378778*378779*378780*378781-1=20584762741360980725159

405753*405754*405755*405756-1=27105247685292348204359
405754*405755*405756*405757-1=27105514894630889556839
405755*405756*405757*405758-1=27105782105945081218679
405756*405757*405758*405759-1=27106049319234932928023

493931*493932*493933*493934-1=59521026444436534848023
493932*493933*493934*493935-1=59521508463394198461239
493933*493934*493935*493936-1=59521990485279505703519
493934*493935*493936*493937-1=59522472510092468429279

510907*510908*510909*510910-1=68135353339657426427639
510908*510909*510910*510911-1=68135886785888068461719
510909*510910*510911*510912-1=68136420235251052702079
510910*510911*510912*510913-1=68136953687746391410559

719310*719311*719312*719313-1=267712108496892516108959
719311*719312*719313*719314-1=267713597213070502749023
719312*719313*719314*719315-1=267715085935457415060959
719313*719314*719315*719316-1=267716574664053270308279

727198*727199*727200*727201-1=279649654061521485614399
727199*727200*727201*727202-1=279651192292672074705599
727200*727201*727202*727203-1=279652730530168510603199
727201*727202*727203*727204-1=279654268774010810760023

790633*790634*790635*790636-1=390753651327250974682919
790634*790635*790636*790637-1=390755628242716568711879
790635*790636*790637*790638-1=390757605165683416667159
790636*790637*790638*790639-1=390759582096151537524023

872674*872675*872676*872677-1=579977484509165175833399
872675*872676*872677*872678-1=579980142901575212869799
872676*872677*872678*872679-1=579982801303124021190023
872677*872678*872679*872680-1=579985459713811621738319

932012*932013*932014*932015-1=754551370432349893400759
932013*932014*932015*932016-1=754554608808552913747679
932014*932015*932016*932017-1=754557847195179746433119
932015*932016*932017*932018-1=754561085592230413825439

***

Jeff wrote:

For puzzle 499, here are the smallest numbers I have found:

to produce 4 primes in a row is 7046
to produce 5 primes in a row is 6985660

None found for 6 primes up to 50.083.988

***

Michael wrote:

got a foursome:

7046*7047*7048*7049-1
7047*7048*7049*7050-1
7048*7049*7050*7051-1
7049*7050*7051*7052-1

***

Frederick wrote:

" <some number>: " indicates the number of numbers multiplied so below : 7046*7047*7048*7049-1, the product of (7047 to 7050)-1, the product of (7048 to 7051)-1 and the product of (7049 to 7052) -1 are all primes. They form a "list of 4".

I only checked through the first million numbers through (a product of) 13 numbers. Only the first solution is listed and in some cases only one was found through 1 million.

4:
List of 4 starting at 7046

5:
List of 4 starting at 634

6:
List of 4 starting at 947
List of 6 starting at 274319 *
List of 5 starting at 508520

7:
List of 4 starting at 60601

8:
List of 4 starting at 60662

9:
List of 4 starting at 111965
List of 5 starting at 381310 *

10:
List of 4 starting at 22604

11:
List of 5 starting at 4 *
List of 4 starting at 362462

12:
List of 4 starting at 55255

13:
List of 4 starting at 347

* : only solution below n=10^6

"Can you find an example of seven prime in a row for some quantity of consecutive integers?" CR

6:
List of 7 prime in arow starting at 75174815

n => n*(n+1)*(n+2)*(n+3)*(n+4)*(n+5)-1 :isprime?
75174815=> 180482166126943940496898850913403914645343699199:1
75174816=> 180482180531941102933063715188668249920708033279:1
75174817=> 180482194936939223469156781227995717406754151759:1
75174818=> 180482209341938302105229028865667883179692507439:1
75174819=> 180482223746938338841331437938000764571002281919:1
75174820=> 180482238151939333677514988283344830221557255999:1
75174821=> 180482252556941286613830659742085000135751680799:1
List ends here, next number is not prime:
75174822=> 1804822669619441976503294321566406457356
26149599:0

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