Problems & Puzzles: Puzzles

Puzzle 986. Consecutive primes from a given set of integers. Gennady Gusev sent the following very nice puzzle:   Find a set of N numbers and expressions with them using the 4 basic arithmetic operations (+, -, *, /) and brackets, and always using the whole set of these numbers, just once each if them, that yield the longest consecutive primes starting in 2.   Gennady sent the following example:   The first 100 primes, from 2 to 541, from the following set of N=5 integers: (2, 8, 16, 19, 21).   2 = 2*8-16-19+21       3 = 2*8/16-19+21       5 = 2+8*(19-16)-21  7 = 2*(8-16+21)-19    11 = (2-8-16)/(19-21)  13 = (2-8*16)/21+19    17 = (2+8)/(16-21)+19  19 = (2*8-16)*21+19    23 = (2-8)/(16-19)+21   29 = 2*(21-8)-16+19    31 = 2*(8+16-19)+21    37 = 2*16+(19+21)/8    41 = 2*8/16+19+21      43 = 2*(8-16+19)+21    47 = 2-8*(16-19)+21    53 = 2*21+8-16+19      59 = 2*(8/16*21+19)    61 = 2*(8/16*19+21)    67 = 2*19-8+16+21      71 = 2*(8+16+21)-19    73 = 2+8-(16-19)*21    79 = 2*8-(16-19)*21    83 = 2*(16+19)-8+21    89 = 2-8-(16-21)*19    97 = 2/8*16*19+21     101 = 2*(16+21)+8+19   103 = 2/8*16*21+19     107 = 2+(8+16-19)*21   109 = 2*(8+16+21)+19   113 = (2-16+19)*21+8   127 = 2/(19-21)+8*16   131 = (2+19)*8-16-21   137 = 2-(8+19)*(16-21)  139 = 8+16/2*19-21     149 = 2+8*19+16-21     151 = 2-(8-16)*21-19   157 = 2-(8-19)*16-21   163 = 2*16+8*19-21     167 = 2+8*21+16-19     173 = 2+8*21-16+19     179 = 8*21-16/2+19     181 = 2*16+8*21-19     191 = (2-8+16)*21-19   193 = (2-21)*(8-19)-16 197 = 2*(8*16-19)-21   199 = 2-(8-19)*16+21   211 = (2-8+16)*19+21   223 = (19-8)*21-16/2   227 = (2+8)*19+16+21   229 = (2-8+16)*21+19   233 = 2+(8-16+19)*21   239 = 2*(8*16-19)+21   241 = (8-2)*(16+21)+19 251 = 2*(8*19-16)-21   257 = (2-8+19)*21-16   263 = 2*16-(8-19)*21   269 = (2-8+21)*19-16   271 = (8+16)/2*21+19   277 = 2-8+16*19-21     281 = (2*16-19)*21+8   283 = 2*8*(19-21/16)   293 = 2+8+16*19-21     307 = (2+19)*16-8-21   311 = 2-8+16*21-19     313 = (2+16)*19-8-21   317 = 2+8*(16+21)+19   331 = 2*16*(19-8)-21   337 = 19*(21-2)-8-16   347 = (8/2+19)*16-21   349 = 2-8+16*21+19     353 = 8-(2-21)*19-16   359 = (2/8+21)*16+19   367 = (2+16)*21+8-19   373 = 2*16*(19-8)+21   379 = (2+21)*16-8+19   383 = 2*(8-16)+19*21   389 = (2+8)*(16+21)+19 397 = 2*16*(21-8)-19   401 = (8/2+16)*19+21   409 = 2-8+16+19*21     419 = 8/2+16+19*21     421 = 8-2+16+19*21     431 = 2*8+16+19*21     433 = (2+19)*21+8-16   439 = 2*16+8+19*21     443 = (2+8+19)*16-21   449 = (2+19)*21-8+16   457 = 2-(8-21)*(16+19) 461 = (2+21)*19+8+16   463 = 8/2*16+19*21     467 = (8/2+19)*21-16   479 = 2+(8+16)*19+21   487 = 2+(8+16)*21-19   491 = (8/2+21)*19+16   499 = (8/2+19)*21+16   503 = (2+16)*(8+21)-19  509 = (8-2+19)*21-16   521 = (8+16)*21-2+19   523 = (2*16-8)*21+19   541 = (2+16)*(8+21)+19 Q1. Is it possible to get the first 100 primes from a "better" set of integers ("better" stands for less than five terms or  for five terms whose sum is smaller than 66). Q2. Send your best solution for the first 1000 primes.

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Contribution came from Gennady Gusev the week 17-24 January, 2020

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Gennady wrote:

This is my solution of Q2: 1003 consecutive primes with the set of 7 numbers:  {...} .See attached file 1003.zip.

N.B. by CR: I will keep veiled the solution sent by Gennady, for the 1003 consecutive primes, for a while or after a first solution comes from another puzzler.

See note below on the Mark Mammel results

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Paul Cleary wrote on Jan 31, 2020

Was unable to find any solutions with a sum of 65 or 64 and incidentally, there are no other solutions than the one published here with sum = 66 with {2, 8, 16, 19, 21}.

 

Here are a few stats I found, with sum = to 66 using {2, 8, 16, 19, 21} and only using {+, -, *, / and ()}.  There are 22352 ways to generate primes. number 2 can be generated the most ways with 2496 different ways and the least is 4 ways with both 347 and 467. The number of different primes generated are 144, the largest being 3139.

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Mark Mammel wrote on Feb 01, 2020:

Attached is my solution for Q2, Puzzle 986 using the set of 7 numbers

set = {2, 5, 6, 7, 8, 9, 56}...

Wow, I found a much smaller solution with

{ 2,6,7,8,9,11,15} attached.

I found equally small sets with:

{4,5,6,7,8,9,19}

and

{5,6,7,8,9,10,13}

Here is his file for the set { 2,6,7,8,9,11,15} which sum up to 58.

BTW, the solution sent last week by Gennady Gusev used a set of 7 integers {12 13 15 17 18 19 20} but with a sum higher, 114. Here is file.

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Gennady checked the solution for Q2 sent by Mark Mammel and found only one mistake:

3089 <> 15 / 6 / 2 * 7 * 8 * 11 + 9 

My correction : 3089 = 15 / 6 * 2 * 7 * 8 * 11 + 9

Other than that I accept that the Mark's solution is better than the mine

Mark replied:

Thanks to Gennady for catching this one -- it looks like I missed needed parenthesis in this case, my output should have been:

3089 = 15 / (6 / 2) * 7 * 8 * 11 + 9
 

My program was using only integer divisions. But of course Gennady's correction is equivalent. I will try to check my list to see if there are any other similar cases where parenthesis are needed. I tried to avoid unneeded parenthesis, but did not take into account that case.

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