Prime Consecutive Triad Sums
- In what follows, I use the term "consecutive triad sums". Here's what I
Take any "X" distinct integers "N" and arrange them in ascending
numerical order N1, N2, N3, N4, N5 N(X). Then the "consecutive triad
sums" are (N1+N2+N3), (N2+N3+N4), (N3+N4+N5) .(N(X-2)+N(X-1)+N(X)).
Obviously for every "X" there are (X-2) triad sums.
Excluding the integer 1, say that we generate a list in ascending
numerical order of all ODD NON-PRIME INTEGERS up to some arbitrary limit.
Here are the first few non-primes (composites) of such a list:
9, 15, 21, 25, 27, 33, 35,
It seems that it is always possible to pick "X"=6 CONSECUTIVE integers
"N" from this list such that the 4 CONSECUTIVE TRIAD SUMS are ALL PRIME,
and that those 4 TRIAD SUMS are ALSO CONSECUTIVE PRIMES. Here is the
largest example I could find via Ubasic:
The 6 CONSECUTIVE NON-PRIME ODD integers are;
43030117, 43030119, 43030125, 43030127, 43030131, 43030133
The resulting 4 PRIME CONSECUTIVE TRIAD SUMS are:
129090361, 129090371, 129090383, 129090391
Now suppose we do the same thing except we use a list of CONSECUTIVE
PRIMES. It seems possible to pick 6 consecutive primes so that the 4
consecutive triad sums are prime, BUT those 4 triad sums will NEVER be
In fact, I will hazard a guess that there is NO number "X" of CONSECUTIVE
PRIMES that one can pick such that their (X-2) consecutive triad sums ARE
CONSECUTIVE primes. Can anyone find a counterexample?
Variations on this theme suggest themselves. I tried a list of ODD
NON-PRIME integers NOT DIVISIBLE BY 3. The largest example I was able get
using "X"=9 consecutive integers from that list is:
The 9 CONSECUTIVE NON-PRIME ODD integers NOT DIVISIBLE by 3 are:
513061, 513605, 513607, 513611, 513613, 513617, 513619, 513623, 513625
The resulting 7 PRIME CONSECUTIVE TRIAD SUMS are:
1540813, 1540823, 1540831, 1540841, 1540849, 1540859, 1540867
In the first example using "X"=6 provides a relatively abundant source of
examples, but I have not found one yet using "X"=7. It might not be
possible. I know this is an utterly useless simple-minded exercise, but
just how far can it be pushed with current computational knowledge?
Ubasic poops out on me if I pick anywhere from "X"=7 to 11 consecutive
integers to process. Thanks folks and regards
- Hi Everybody
Correction to my post "Prime Consecutive Triad Sums", Yahoo message
13449. There I wrote " In fact, I will hazard a guess that there is NO
number "X" of CONSECUTIVE PRIMES that one can pick such that their (X-2)
consecutive triad sums ARE CONSECUTIVE primes. Can anyone find a
counterexample?". I found 2 counterexamples for X=5 and probably there
are some really huge ones lurking in outer prime space. Bet a K-Tuple
hunter could nail one.
These 5 CONSECUTIVE primes 61, 67, 71, 73, 79 will yield these 3
consecutive triad sums 199, 211, 223 which ARE CONSECUTIVE primes.
These 5 CONSECUTIVE primes 2683, 2687, 2689, 2693, 2699 will yield these
3 consecutive triad sums 8059, 8069, 8081 which ARE CONSECUTIVE primes.
My humble apology.