A271725 T(n,k) is an array read by rows, with n > 0 and k=1..4, where row n gives four prime numbers in increasing order with locations in right angles of each concentric square drawn on a distorted version of the Ulam spiral.

3, 7, 17, 19, 13, 23, 37, 41, 307, 359, 401, 419, 13807, 14159, 14401, 14519, 41413, 42023, 42437, 42641, 6317683, 6325223, 6330257, 6332771, 22958473, 22972847, 22982437, 22987229, 39081253, 39100007, 39112517, 39118769, 110617807, 110649359, 110670401, 110680919

Offset: 1

Author

Michel Lagneau, Apr 13 2016

Keywords

• nonn

Comments

See the illustration for more information.

Conjecture: there is an infinity of concentric squares having a prime number in each right angle. The number 5 is the center of all the squares.

It seems that the drawing of an infinite number of concentric squares having a prime number in each corner is impossible in an Ulam spiral. But with a slight distortion of this space, the problem becomes possible.

The illustration (see the link) shows the new version of a spiral with two remarkable orthogonal diagonals containing four classes of prime numbers given by the sequences A125202, A121326, A028871 and A073337 supported by four line segments. These intersect at a single point represented by the prime number 5.

The sequence of the corresponding length of the sides is {s(k)} = {2, 4, 18, 118, 204, 2514, 4792, 6252, 10518, 14032, 16752, 17598, ...}

The primes are defined by the polynomials: [4*m^2-10*m+7, (2*m-1)^2-2, 4*m^2+1, 4*(m+1)^2-6*(m+1)+1]. The sequence of the corresponding m is {b(k)} = {2, 3, 10, 60, 103, 1258, 2397, 3127, 5260, 7017, 8377, 8800, 10375, 11518, 11523, 12498, 15415, 15888, ...} with the relation b(k) = 1 + s(k)/2.

The array begins:

3, 7, 17, 19;

13, 23, 37, 41;

307, 359, 401, 419;

13807, 14159, 14401, 14519;

41413, 42023, 42437, 42641;

...

Construction of the spiral (see the illustration in the link):

. . . . . . . . . . . .

. 42 41 40 39 38 37 . . .

|

. 43 20 19 18 17 36 35 . .

|

. . 21 6 5 16 15 34 . .

|

. . 22 7 4 3 14 33 . .

. . 23 8 1 2 13 32 . .

. . 24 9 10 11 12 31 . .

. . 25 26 27 28 29 30 . .

. . . . . . . . . . .

The first squares of center 5 having a prime number in each vertex are:

19 18 17 41 40 39 38 37

6 5 16 20 19 18 17 36

7 4 3 21 6 5 16 15 . . . .

22 7 4 3 14

23 8 1 2 13

Links

• Michel Lagneau, Illustration

Crossrefs

Cf. A028871, A073337, A121326, A125202, A200975.

Programs

Maple

for n from 1 to 10000 do :
  x1:=4*n^2-10*n+7:x2:=(2*n-1)^2-2:
  x3:=4*(n+1)^2-6*(n+1)+1:x4:=4*n^2+1:
   if isprime(x1) and isprime(x2) and isprime(x3) and isprime(x4)
    then
     printf("%d %d %d %d %d \n",n,x1,x2,x4,x3):
    else
    fi:
od: