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Primes forming a 3 X 3 magic square with prime entries and minimal constant 177 = A164843(3).
+20
8
5, 17, 29, 47, 59, 71, 89, 101, 113
COMMENTS
Sequence A073473 gives a variant using "primes including 1" (for historical reasons), to which also refers A073502. - M. F. Hasler, Oct 24 2018
EXAMPLE
The square is [101 5 71 ; 29 59 89 ; 47 113 17].
The lexicographically smallest equivalent variant (modulo reflections on the symmetry axes of the square) is [17 89 71 ; 113 59 5 ; 47 29 101], cf. A320872. - M. F. Hasler, Oct 24 2018
PROG
(PARI) A024351=select(p->setsearch(P, 118-p), P=primes(30)[^5]) \\ 118 = 2*59, where 59 is the central prime; primes(30) = primes < 118. For the magic square itself, use A320872_row(1). - M. F. Hasler, Oct 25 2018
CROSSREFS
Cf. A320872 (3 X 3 magic squares of primes), A268790 (magic sums of these).
AUTHOR
Karl Schmerbauch (karl.j.schmerbauch(AT)boeing.com)
Magic sums of 3 X 3 magic squares composed of primes.
+10
7
177, 213, 219, 267, 309, 327, 381, 393, 411, 417, 447, 453, 471, 501, 519, 537, 573, 579, 633, 681, 717, 723, 753, 771, 789, 807, 813, 843, 849, 879, 921, 933, 1011, 1041, 1047, 1059, 1077, 1101, 1119, 1137, 1149, 1167, 1191, 1203, 1227, 1257, 1263, 1293
COMMENTS
All terms are 3 times odd primes.
3*p is a term if and only if p is a prime not in A073350. Conjecture: 3*p is a term for every prime > 859.
I verified this for all primes < 100000.
The Green-Tao theorem implies the sequence is infinite: given one magic square with entries a(i,j), there are infinitely many pairs of positive integers x,y such that b(i,j) = x + y*a(i,j) are all prime. Then b(i,j) form another magic square. (End)
The terms equal three times the central elements (and equivalently, one third of the sum of all elements) of the 3 X 3 magic squares made of primes, which are listed in A320872. - M. F. Hasler, Oct 28 2018
FORMULA
If conjecture is true, a(n) = 3*prime(n+40) for n >= 110. - Robert Israel, Feb 16 2016
EXAMPLE
Examples of 3 X 3 magic squares composed of primes.
.
+---+---+---+
| 17| 89| 71|
+---+---+---+
|113| 59| 5 |
+---+---+---+
| 47| 29|101|
+---+---+---+
The magic constant is 177 = a(1).
.
+---+---+---+
| 41| 89| 83|
+---+---+---+
|113| 71| 29|
+---+---+---+
| 59| 53|101|
+---+---+---+
The magic constant is 213 = a(2).
MAPLE
N:= 10000: # to get all terms <= N P:= select(isprime, {seq(p, p=3..2*N/3, 2)}):
count:= 0:
for ic from 1 while P[ic] <= N/3 do
c:= P[ic];
V:= map(`-`, P[ic+1..-1], c) intersect map(t -> c-t, P[1..ic-1]);
nv:= nops(V);
VV:= {seq(seq(V[j]-V[i], j=i+1..nv), i=1..nv-1)} intersect V;
nvv:= nops(VV);
found:= false;
for ia from 1 to nvv while not found do
a:= VV[ia];
for ib from ia+1 to nvv while VV[ib] < c - a do
b:= VV[ib];
if b <> 2*a and {c-a-b, c-a+b, c-b+a, c+a+b} subset P then
found:= true;
count:= count+1;
A[count]:= 3*c;
break
fi
od
od:
od:
PROG
(PARI) c=3; A268790_vec=3*vector(50, i, c= A320872_row(1, 0, c+1)[2, 2]) \\ Illustrates formula & comment. - M. F. Hasler, Oct 28 2018 (PARI) is_ A268790(c)={denominator(c/=3)==1&& isprime(c)&& forstep(a=2, c\2-1, 2, isprime(c-a)&& isprime(c+a)&& forstep(b=2, c-2*a-2, 2, isprime(c-2*a-b)&& isprime(c-a-b)&& isprime(c-b)&& isprime(c+b)&& isprime(c+a+b)&& isprime(c+2*a+b)&& return(1)))} \\ M. F. Hasler, Oct 28 2018
The smallest magic constant for n X n magic square with prime entries (regarding 1 as a prime).
+10
5
111, 102, 213, 408, 699, 1114, 1681, 2416, 3355, 4514, 5937, 7626, 9635, 11986, 14691, 17818, 21373, 25394, 29873, 34926, 40511, 46664, 53445, 60898, 69045, 77888, 87473, 97850, 109065, 121126, 134113, 147982, 162759
COMMENTS
Until the early part of the twentieth century 1 was regarded as a prime (see A008578).
REFERENCES
W. S. Andrews and H. A. Sayles, The Monist (Chicago) for October 1913.
H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 125, who quotes the Andrews-Sayles article as his source.
EXTENSIONS
Dudeney gives 36095/11 for n = 11 (an obvious typo) and 4514 for n = 12
a(3)-a(12) are confirmed/given by Chebrakov
a(15), a(17), a(22), a(35), and a(124)=9912840 from S. Tognon (cf. A173079) a(13)-a(14), a(16), a(18)-a(21), a(23)-a(34) from N. Makarova
Primes (including 1) forming 3 X 3 magic square with prime entries and minimal constant 111 = A073502(3).
+10
3
1, 7, 13, 31, 37, 43, 61, 67, 73
COMMENTS
Until the early part of the twentieth century 1 was regarded as a prime (cf. A008578). "The problem of constructing magic squares with prime numbers only was first discussed by myself in The Weekly Dispatch for Jul 22 1900 and Aug 05 1900; but during the last three or four years it has received great attention from American mathematicians. First, they have sought to form these squares with the smallest possible constants.
"Thus the first nine prime numbers, 1 to 23 inclusive, sum to 99, which (being divisible by 3) is theoretically a suitable series; yet it has been demonstrated that the smallest possible constant is 111 and the required series as follows: 1,7,13,31,37,43,61,67,73." - Dudeney
See A024351 for the "modern" version of the minimal 3 X 3 magic square of primes. - M. F. Hasler, Oct 30 2018
REFERENCES
H. E. Dudeney, Amusements in Mathematics, Nelson, London, 1917, page 125.
EXAMPLE
The square is [ 43 1 67 / 61 37 13 / 7 73 31 ].
AUTHOR
Lee Sallows (Sallows(AT)psych.kun.nl), Aug 27 2002
Magic sums of 4 X 4 magic squares composed of primes.
+10
3
120, 126, 132, 136, 138, 140, 142, 144, 146, 148, 150, 154, 156, 158, 160, 162, 164, 166, 168, 170, 172, 174, 176, 178, 180, 182, 184, 186, 188, 190, 192, 194, 196, 198, 200, 202, 204, 206, 208, 210, 212, 214, 216, 218, 220, 222, 224, 226, 228, 230, 232, 234
COMMENTS
I conjecture that every even number greater than 152 belongs to this sequence.
FORMULA
If conjecture is true, a(n) = A005843(n+65) for n >= 12.
EXAMPLE
Examples of 4 X 4 magic squares composed of primes.
.
|---|---|---|---|
| 3 | 7 | 43| 67|
|---|---|---|---|
| 31| 61| 17| 11|
|---|---|---|---|
| 73| 23| 19| 5 |
|---|---|---|---|
| 13| 29| 41| 37|
|---|---|---|---|
The magic constant is 120 = a(1).
.
|---|---|---|---|
| 3 | 7 | 43| 73|
|---|---|---|---|
| 31| 67| 17| 11|
|---|---|---|---|
| 79| 23| 19| 5 |
|---|---|---|---|
| 13| 29| 47| 37|
|---|---|---|---|
The magic constant is 126 = a(2).
Positive integers n such that the sum S of 1 and first n^2-1 odd primes is divisible by n and S/n == n (mod 2).
+10
2
1, 2, 3, 12, 15, 17, 22, 35, 124, 191, 774, 1405, 1522, 3988, 6220, 7448, 8038, 11404, 63027, 161153, 582096
COMMENTS
A necessary condition for the existence of n X n magic square consisting of 1 and the first n^2-1 odd primes.
In 1913, J. N. Muncey proved that 12 is actually the smallest (nontrivial) order for which such a magic square exists.
Squares of order 15, 17, 22, 35 and 124 were constructed by S. Tognon.
The number S/n, if it exists, is also called the potential magic constant.
It is believed that the corresponding magic squares do exist for any order a(n) with n >= 4. (End)
EXAMPLE
The case a(1) = 1 is trivial.
In case a(2) = 2, the set of potential magic square numbers is {1, 3, 5, 7} with potential magic constant 8, however, no magic square exists of order 2.
In case a(4) = 12, not only the potential magic constant exists, but also the magic square itself, as shown by Stefano Tognon or Eric Weisstein's World of Mathematics. (End)
The smallest constant of an n X n associative magic square composed of distinct primes.
+10
2
177, 240, 1255, 630, 4487, 2040, 12249
COMMENTS
In the associative magic square the sums of every pair of elements which are center-symmetrical are equal. Associative squares exist for every magic square order.
a(10) <= 4950, a(11) = 26521, a(12)=8820, a(13)<=50453, a(14)=16170, a(15)=74595, a(16)<=25200, a(17)=128197, a(18)=35910, a(19)<=193363, a(20)=54600.
EXAMPLE
a(7) = 4487:
53 1277 101 1091 173 1019 773
1013 59 863 599 881 1049 23
179 1193 563 821 761 131 839
1031 311 929 641 353 971 251
443 1151 521 461 719 89 1103
1259 233 401 683 419 1223 269
509 263 1109 191 1181 5 1229
a(8) = 2040:
7 499 19 487 463 67 467 31
53 421 233 409 317 157 379 71
61 347 239 401 313 227 373 79
173 311 241 179 383 281 359 113
397 151 229 127 331 269 199 337
431 137 283 197 109 271 163 449
439 131 353 193 101 277 89 457
479 43 443 47 23 491 11 503
a(9) = 12249:
1283 311 1811 2213 1571 569 2039 1163 1289
773 653 2243 1619 2063 593 2693 383 1229
1979 1499 2699 641 821 89 809 2003 1709
1613 2531 101 131 2333 2441 2663 263 173
113 179 2711 449 1361 2273 11 2543 2609
2549 2459 59 281 389 2591 2621 191 1109
1013 719 1913 2633 1901 2081 23 1223 743
1493 2339 29 2129 659 1103 479 2069 1949
1433 1559 683 2153 1151 509 911 2411 1439
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