Problems & Puzzles: Puzzles

 Puzzle 502. Adding consecutive 2n+1 primes a prime JM Bergot sent the following puzzle related to adding 2n+1 consecutive primes: 31 = 31 31+37+41=109 109+113+127+131+137=617 617+619+631+641+643+647+653=4451 4451+4457+4463+4481+4483+4493+4507+4513+4519=40367 (FAIL) Q. Can you find a larger pyramid like the one above?

Contributions came from J. K. Andersen, Jan van Delden, Farideh Firoozbakht, Seiji Tomita, Robin Garcia,
Dimitry Kamenetsky, Luca Poletti, Frederick Schneider, Emmanuel Vantieghem.

By the way Dimitry, Luca & Emmanuel are invited to send their respective short bios to be added to the The Puzzlers page.

Of course we expect too they send once in a while prime-puzzles for these pages.

JKA wrote:

37330116097 starts a pyramid with 10 prime sums:

37330116097 = 37330116097
37330116097 + ... + 37330116149 = 111990348347
111990348347 + ... + 111990348443 = 559951741967
559951741967 + ... + 559951742171 = 3919662194621
3919662194621 + ... + 3919662194999 = 35276959752709
35276959752709 + ... + 35276959753027 = 388046557281451
388046557281451 + ... + 388046557281809 = 5044605244660991
5044605244660991 + ... + 5044605244661329 = 75669078669917287
75669078669917287 + ... + 75669078669917753 = 1286374337388597683
1286374337388597683 + ... + 1286374337388598399 = 24441112410383362873

The full 10th line would have 400 characters without spaces.
The 11th sum is the composite 513263360618050629815.

***

Jan wrote:

I found the following (first occurring) solutions [n,p]:

1 5
2 7
3 31
4 431
5 463
6 5715817
7 36943

For every p, the final sum to test given n, (length 2n+1), is at least (2n+3).(2n+1)....5.3.p, which is equal
to (2n+3)!/[(n+1)!*2^(n+1)].p. The order of growth is approximately: (2.(n+1)/e)^(n+2).p.
I searched with p<=10^9. Luckily the average value of n is small in this range (0.135).

***

Farideh wrote:

For the prime 36943 the last summation which produce a prime has 15 summands.

36943 = 36943
36943+36947+36973 = 110863
110863+110879+110881+110899+110909 = 554431
554431+554447+554453+554467+554503+554527+554531 = 3881359
...
...
...

4995384923+4995384929+4995384931+4995384997+4995385007+4995385013+
4995385027+4995385033+4995385061+4995385063+4995385103+4995385139+
4995385181+4995385211+4995385223 = 74930775841

74930775841+74930775851+74930775889+74930775919+74930775923+
74930775943+74930775947+74930775967+74930775979+74930776049+
74930776093+74930776153+74930776183+74930776211+74930776231+
74930776267+74930776279 = 1273823192725 (FAIL)

***

Seiji wrote:

Consecutive 2n+1 primes:

n= 0 p1
n= 1 p1+p12+p13=p2
n= 2 p2+p22+p23+p24+p25=p3
n= 3 p3+p32+p33+p34+p35+p36+p37=p4
n= 4 p4+p42+p43+p44+p45+p46+p47+p48+p49=p5
n= 5 p5+p52+p53+p54+p55+p56+p57+p58+p59+p5a+p5b=p6
n= 6 p6+p62+p63+p64+p65+p66+p67+p68+p69+p6a+p6b+p6c+p6d=p7
n= 7 p7+p72+p73+p74+p75+p76+p77+p78+p79+p7a+p7b+p7c+p7d+p7e+p7f=p8

Search condition: p1 < 10^10, 4 <= n <= 10

The smallest solution is

Case of n=4: p1 = 431
Case of n=5: p1 = 463
Case of n=6: p1 = 36943( 2nd: 5715817 )
Case of n=7: p1 = 36943

I could not find the solution for the case of n > 7.

***

Robin wrote:

4 steps

431, 439-->1303
1303, 1327-->6577
6577, 6653--> 46273
46273, 46351-->416833

5 steps

463, 479-->1409
1409, 1433--> 7121
17121, 7187-->50051
50051, 50119-->450761
450761, 450839-->4958819

6 and 7 steps

36943, 36973-->
110863, 110909-->
554431, 554531-->
3881359, 3881441-->
34932613, 34932851-->
384260249, 384260479-->
4995384923, 4995385223-->74930775841

***

Dimitry wrote:

I have found a pyramid with 6 levels:
463 = 463
463+467+479 = 1409
1409+1423+1427+1429+1433 = 7121
7121+7127+7129+7151+7159+7177+7187 = 50051
50051+50053+50069+50077+50087+50093+50101+50111+50119 = 450761
450761+450767+450787+450797+450799+450803+450809+450811+450817+450829+450839 = 4958819

No larger pyramid exists if the starting prime is less than 10000.

***

Luca Poletti wrote:

Up to now the longevousest base for the pyramid I found is:

36943

which fails at 1273823192725 after 8 passages.

another wich fails after 8 passages is 588083233 with a sum of 20265013128025579
(not fully sure about that, I used a miller rabin probabilistic test so there are 4^(-7) possibility of error in that case)

I checked up to 653607433.

***

Fred wrote:

Here are the minimal answers through 8. There is no larger pyramid beginning from under 3 billion.

Pyramid started from 3 of length: 1
3=3

Pyramid started from 5 of length: 2
5=5
5+7+11=23

Pyramid started from 7 of length: 3
7=7
7+11+13=31
31+37+41+43+47=199

Pyramid started from 31 of length: 4
31=1
31+37+41=109
109+113+127+131+137=617
617+619+631+641+643+647+653=4451

Pyramid started from 431 of length: 5
431=431
431+433+439=1303
1303+1307+1319+1321+1327=6577
6577+6581+6599+6607+6619+6637+6653=46273
46273+46279+46301+46307+46309+46327+46337+46349+46351=416833

Pyramid started from 463 of length: 6
463=463
463+467+479=1409
1409+1423+1427+1429+1433=7121
7121+7127+7129+7151+7159+7177+7187=50051
50051+50053+50069+50077+50087+50093+50101+50111+50119=450761
450761+450767+450787+450797+450799+450803+450809+450811+450817+450829+450839=4958819

Pyramid started from 5715817 of length: 7
5715817=5715817
5715817+5715823+5715839=17147479
17147479+17147483+17147491+17147513+17147531=85737497
85737497+85737503+85737521+85737539+85737577+85737593+85737611=600162841
600162841+600162869+600162931+600162949+600162967+600162991+600163027+600163049+
600163063=5401466687
5401466687+5401466701+5401466711+5401466713+5401466767+5401466803+5401466827+
5401466879+5401466951+5401466953+5401466957=59416134949

59416134949+59416134977+59416135007+59416135021+59416135099+59416135111+59416135117+
59416135153+59416135217+59416135231+59416135249+59416135357+59416135391=772409756879

Pyramid started from 36943 of length: 8
36943=36943
36943+36947+36973=110863
110863+110879+110881+110899+110909=554431
554431+554447+554453+554467+554503+554527+554531=3881359

3881359+3881363+3881387+3881393+3881407+3881413+3881419+3881431+3881441=34932613

34932613+34932673+34932683+34932701+34932719+34932757+34932761+34932823+34932829
+34932839+34932851=384260249

384260249+384260273+384260309+384260321+384260347+384260363+384260399+384260413+
384260419+384260431+384260447+384260473+384260479=4995384923

4995384923+4995384929+4995384931+4995384997+4995385007+4995385013+4995385027+4995385033+
4995385061+4995385063+4995385103+4995385139+4995385181+4995385211+4995385223=74930775841

***

Emmanuel wrote:

I was able to find an 8-line pyramid :

36943, prime
36943+36947+36973 = 110863, prime
110863+110879+110881+110899+110909 = 554431, prime
554431+554447+554453+554467+554503+554527+554531 = 3881359, prime
3881359+3881363+3881387+3881393+3881407+3881413+3881419+3881431+3881441 = 34932613, prime
34932613+34932673+34932683+34932701+34932719+34932757+34932761+34932823+34932829+34932839+34932851 = 384260249, prime
384260249+384260273+384260309+384260321+384260347+384260363+384260399+384260413+384260419+384260431+384260447+384260473+384260479 = 4995384923, prime
4995384923+4995384929+4995384931+4995384997+4995385007+4995385013+4995385027+4995385033+4995385061+4995385063+4995385103+4995385139+4995385181+4995385211+4995385223 = 74930775841, prime.

Any other solution with a number of lines > 7 should have as top a prime number bigger than the 10milion-th prime.

***

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