now, mathematicians only have considered lineal constellations
of primes (The called K-Tuples ) It is notable the work of
Tony Forbes who have searched for the largest number of primes
contained in a given interval of integers. See:
example the maximum number of primes in an interval of 16 is 6.
In an interval of 36 is 11 and in an interval of 50 is 14.
cast the conjecture : “There is not interval N, N+x that
contains more primes that the first interval 0 , x.” (This
conjecture is almost certainly false)
puzzle I will introduce constellations in two dimensions..
it is necessary to adopt a standard matrix for the Eratosthenes’
the sieve of 210 columns, but discarded the columns containing
only composite numbers ,the rest are the 48 that can contain
of manage easily this game in a computer, I took as standard
plane a net of cells in 48 columns x 33 rows. That spans an
interval of 6930 integers.
row is constituted by the first 48 numbers not divisible by
that 121, 143, 169, 187 & 209 are not primes.)
horizontal differences between cells in the rows are
2, 4, 2,
4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2,
4, 8, 6, 4, 6, 2, 4, 6 , 2 ,6
6, 4, 2,
4, 6, 2, 6, 4, 2, 4, 2, 10, 2, 10
given interval is <= 210 , it is in this chain of differences
where we must search for the maximum possible number of
primes. For example if the interval is 16, the optimum number of
primes will be in the minimum chunk of the sequence that sums
16, that is : 4, 2, 4, 2, 4 . The primes are:
p+6, p+10, p+12, p+16 Example : 97, 101, 103, 107, 109, 113 .
vertical columns form arithmetical progressions with 210 of
build in a computer’s screen, a matrix 48x33 of little squares
separated by narrow spaces and the squares corresponding to
prime numbers are colored, and the composite numbers are
invisible, then we can see the constellations of primes..
is to discover isolated symmetrical or beautiful patterns and
to investigate the cases that are impossible to appear. (Only
the pages after the first are considered.)
To make a
program that realize the described matrix. Stopping at each
screen and signaling the number of page. The input for the
beginning will be the number of the page or the integer we want
to initiate the search.
1.- In what page is a
chain of 12 consecutive primes?
2.- To find an isolated
compact rectangle of 4x3 primes. Or a box 5x3 primes
3.- It’s possible to
find a column with 12 adjacent primes?
4.- It’s possible a row
full of illuminated primes?.