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?

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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.

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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).

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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.

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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.

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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.

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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

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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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