Problems & Puzzles: Puzzles

Puzzle 567.- p1+...+pk = q1*...*qk

JM Bergot sent the following puzzle:

One sees that

33,263=29*31*37=11,083+11,087+11,093.

The product of three consecutive primes is equal to the sum of three consecutive primes.

Q. Can you find such numbers having five or more consecutive primes in both the sum and the product? (Please just send your largest quantity of members example)


Contributions came from: J. K. Andersen, Jerrold W. Pease, Jan van Delden, Seiji Tomita, Antoine Verroken, Emmanuel Vantieghem

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J. K. Andersen wrote:

In 2003 in puzzle 222 I found the smallest solution for 37 primes:
23663*...*23993 = (23993#/23633#-29)/37-6601 + ... + (23993#/23633#-29)/37+5887

A solution with 55 proven primes: 16447 * ... * 16993 = 16993#/16433#
= (16993#/16433#-47)/55-12273 + ... + (16993#/16433#-47)/55+14775

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Jerrold W. Pease wrote:

I found this example with 7-factors / 7-addends:
 
 
   229 * 233 * 239 * 241 * 251 * 257 * 263
 
   = 7448535640735789 + 7448535640735843 +
      7448535640735867 + 7448535640735877 +
      7448535640735991 + 7448535640736009 +
      7448535640736087
 
   = 52139749485151463

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Jan van Delden wrote:

My largest solution (27 terms):
 
1278027407299563770535135132259390872884161116614603576749845345793233055219143
6057459407845862111339342001301009074914115233=p[4143]* ..*p[4169]=39659#/39343#
and the sum of:
47334348418502361871671671565162624921635596911651984324068346140490113156264577990
5903994291189308864518566704039811630439+..+
47334348418502361871671671565162624921635596911651984324068346140490113156264577990
5903994291189308864518566704039811637763

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Seiji Tomita wrote:

67* 71* 73* 79* 83* 89* 97* 101* 103* 107* 109 = 216819892656221844131
                                              + 216819892656221844133
                                              + 216819892656221844139
                                              + 216819892656221844169
                                              + 216819892656221844307
                                              + 216819892656221844331
                                              + 216819892656221844347
                                              + 216819892656221844373
                                              + 216819892656221844397
                                              + 216819892656221844401
                                              + 216819892656221844421
 

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Antoine Verroken wrote:

Please will find below solution to p.567 :11 consecuive primes :

Sum,product :50654724288442467111896354027508920402535323 
Product : 9349 * 9371 * * 9433
Sum     : 4604974935312951555626941275228083672957249 + ... +
4604974935312951555626941275228083672958037 

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Emmanuel Vantieghem wrote:

I found solutions for every odd  n <= 25.  The biggest I could find was :

11783*11789*11801*11807*...*11941*11953*11959*11969*11971 = 296335465676181118786625015092344431506165371405957047466285893482841032230573
98662776857001628028993+
...

+296335465676181118786625015092344431506165371405957047466285893482841032230573
98662776857001628033813
All the primes are proved primes (I used Mathematica).

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