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A320874 Lexicographically first 4 X 4 pandiagonal magic square made of consecutive primes.

%I A320874 #24 Apr 17 2022 03:54:58
%S A320874 170693941183817,170693941183933,170693941183949,170693941183981,
%T A320874 170693941183979,170693941183951,170693941183847,170693941183903,
%U A320874 170693941183891,170693941183859,170693941184023,170693941183907,170693941183993,170693941183937,170693941183861,170693941183889
%N A320874 Lexicographically first 4 X 4 pandiagonal magic square made of consecutive primes.
%C A320874 This is also the 4 X 4 pandiagonal magic square made of consecutive primes which has the smallest possible magic constant (= sum), 682775764735680 = A256234(1). (In the present case there is no other non-equivalent pandiagonal 4 X 4 magic square having the same magic sum, but this could be possible as for rows 7 and 8 of A320872.)
%C A320874 There exist many non-pandiagonal 4 X 4 magic squares composed of consecutive primes with much smaller magic constant, the smallest being A073520(4) = 258.
%C A320874 Pandiagonal means that not only the 2 main diagonals, but also the 6 other "broken" diagonals all have the same sum, Sum_{i=1..4} A[i,M4(k +- i)] = 682775764735680 for k = 1, ..., 4 and M4(x) = y in {1, ..., 4} such that y == x (mod 4).
%C A320874 A pandiagonal magic square allows rotations (but not arbitrary cyclic permutations like, e.g., 1 -> 3 -> 4 -> 1) of columns or rows, as well as reflection on the 4 symmetry axes of the square (which also produce rotations of 90 degrees around the center of the square). Among all these variants of this square, there is none with elements coming earlier than (170693941183817, 170693941183933, ...), cf. PROGRAM for explicit verification.
%C A320874 The same 4 X 4 primes are given in increasing order in sequence A245721. But does not give more information than smallest term, the central term, or the magic constant itself (cf. A256234) which uniquely determines the sequence of primes (cf. PARI code) since they have to be consecutive and their sum is equal to 4 times the magic constant. The present sequence gives the full information about the magic square, and the given PARI code allows the production of all "equivalent" variants of the square.
%D A320874 Allan W. Johnson, Jr., Journal of Recreational Mathematics, vol. 23:3, 1991, pp. 190-191.
%D A320874 Clifford A. Pickover, The Zen of Magic Squares, Circles and Stars: An Exhibition of Surprising Structures across Dimensions, Princeton University Press, 2002.
%H A320874 Harvey Heinz, <a href="http://www.magic-squares.net/primesqr.htm">Prime Magic Squares</a>
%H A320874 <a href="/index/Mag#magic">Index entries for sequences related to magic squares</a>
%e A320874 The magic square is
%e A320874   [ 170693941183817 170693941183933 170693941183949 170693941183981 ]
%e A320874   [ 170693941183979 170693941183951 170693941183847 170693941183903 ]
%e A320874   [ 170693941183891 170693941183859 170693941184023 170693941183907 ]
%e A320874   [ 170693941183993 170693941183937 170693941183861 170693941183889 ]
%o A320874 (PARI) /* the following transformation operators for matrices, together with transposition, allow the production of all (24 for n=4) variants of a (pandiagonal) magic square */
%o A320874 REV(M)=matconcat(Vecrev(M)) \\ reverse the order of columns of M
%o A320874 FLIP(M)=matconcat(Colrev(M)) \\ reverse the order of rows of M
%o A320874 ROT(M,k=1)=matconcat([M[,k+1..#M],M[,1..k]]) \\ rotate left by k (default: 1) columns
%o A320874 ALL(M)=Set(concat(apply(M->vector(#M,k,ROT(M,k)),[M,M~,REV(M),REV(M~),FLIP(M),FLIP(M~)]))) \\ PARI orders the set according to the (first) columns of the matrices, so one must take the transpose to get them ordered according to elements of the first row.
%o A320874 \\ The set of primes is A245721=MagicPrimes(682775764735680,4), cf. A073519.
%Y A320874 Cf. A073519 and A320873, A073521, A073522 (3 X 3, 4 X 4 and 5 X 5 consecutive primes), A073523 and A320876 (6 X 6 consecutive primes, pandiagonal magic square).
%Y A320874 Cf. A210710: Minimal index of a Stanley antimagic square of order n consisting of distinct primes.
%Y A320874 Cf. A073520: Smallest magic sum for an n^2 magic square made of consecutive primes.
%Y A320874 Cf. A104157: Smallest of n X n consecutive primes forming a magic square.
%Y A320874 Cf. A256234: Magic sums of 4 X 4 pandiagonal magic squares of consecutive primes.
%K A320874 nonn,fini,full
%O A320874 1,1
%A A320874 _M. F. Hasler_, Oct 22 2018