In this post I want to show a few examples of how a prime number can be obtained from all the preceding prime numbers and 1 using addition and subtraction. In Part 1 I’ll deal with only even-indexed primes: a(n)=prime(2n). 3 is the first even-indexed prime, 7 is the second, 13 is the third, 19 is the fourth and so on. See the OEIS sequence A031215 for more details.
The idea for this post is due to Curiosa 67 by Renato Della Torre in Scripta Mathematica ( page 159, Volume 7, 1940). You can use this link to see a PDF version of volume 7 from a library in Romania.
Examples
For convenience I included a snapshot of Curiosa 67 below so you can see the original examples.
For convenience I will include the examples from Curiosa 67 again below, plus adding a few new examples. The summation for 19 was wrong (since it included the non-prime 9 instead of 7) , so I corrected it. The numbers in the summation are ordered in increasing order (according to their absolute value):
p(2)=3=1+2
p(4)=7=1-2+3+5
p(6)=13=1+2-3-5+7+11
p(8)=19=1+2+3+5−7+11−13+17
p(10)=29=−1−2−3+5−7+11+13+17+19−23
p(12)=37=−1−2−3−5−7+11−13+17+19+23+29−31
p(14)=43= −1−2−3−5−7−11−13+17−19+23+29+31−37+41
p(16)=53= −1−2−3−5−7−11−13−17−19−23+29+31−37+41+43+47
p(18)=61=−1−2−3−5−7−11−13−17+19+23+29−31+37+41+43−47−53+59
p(20)=71=−1−2−3−5−7−11−13−17−19−23−29−31+37−41+43−47+53+59+61+67
p(22)=79=−1−2−3−5−7−11−13−17−19−23−29+31+37+41+43+47+53−59−61−67+71+73
p(24)=89=−1−2−3−5−7−11−13−17−19−23−29−31−37+41−43+47+53+59+61+67+71−73−79+83
p(26)=101=−1−2−3−5−7−11−13−17−19−23−29−31−37−41+43+47+53+59+61+67+71−73−79+83−89+97
p(28)=107=−1−2−3−5−7−11−13−17−19−23−29−31−37−41−43−47−53−59+61−67−71+73+79+83+89+97+101+103
p(30)=113=−1−2−3−5−7−11−13−17−19−23−29−31−37−41−43−47−53−59−61+67+71+73−79+83+89+97+101−103+107+109
Discussion
I used ChatGPT with Wolfram API to obtain the summations not included in Curiosa 67. ChatGPT had no problem with the primes up to 53 (once it understood exactly what type of summation I want). Starting with 61, ChatGPT required some refinements in the query. For example, for 61 and above I always guessed that the last term in the summation (biggest prime) is positive. For the bigger primes I also guessed that the first few terms should be negative.
Are these summations always unique? It will also be interesting to know if the pattern of negative initial terms and/or positive last term is still present for larger primes.
Addendum: OEIS Sequence A113040
The problem was tackled by the OEIS sequence A113040. The sequence shows the number of solutions to each summation, so we know that the solutions is not usually unique. The solutions are unique for 3 and 7, but 13 has 3 solutions. For higher prime numbers the number of solutions increase rapidly.
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