Magic squares of twin primes

[Natalia Makarova]

Magic squares of twin primes

See
http://www.primepuzzles.net/puzzles/puzz_080.htm

n=3 (minimal, author Radko Nachev)

239 17  191
101 149 197
107 281 59

S=447

Magic square of the second number-twins (+2):

241 19  193
103 151 199
109 283 61

S=453

I found the following solutions:

n=4 (minimal)

17  11  419 137
269 227 59  29
107 197 101 179
191 149 5   239

S=584

n=5 (minimal)

5   11  617 179 281
71  311 101 191 419
239 461 149 17  227
347 41  29  569 107
431 269 197 137 59

S=1093

n = 6 (minimal)

5    17  1049 11  857 419 
29   881 41   59  521 827 
149  659 569  227 137 617 
347  431 281  641 461 197 
1019 101 239  821 71  107 
809  269 179  599 311 191

S=2358

I have not found a solution for n = 7.
Next, I already have a solution for n = 8 (minimal), n = 9 (not minimal ?), n = 10 (minimal).

Anybody can find a solution for n = 7?

I found pandiagonal square 7th order of primes twins.
See http://www.primepuzzles.net/puzzles/puzz_689.htm

But this magic square has a very large magic constant.

[Natalia Makarova]

My solutions to the puzzle # 80

n=8 (minimal)

179, 419, 1277, 239, 1721, 1451, 71, 1997,
1229, 821, 599, 1667, 191, 1319, 1301, 227,
107, 2141, 41, 2027, 281, 809, 1931, 17,
29, 1289, 1487, 101, 641, 1787, 149, 1871,
1949, 461, 1619, 857, 1091, 311, 5, 1061,
2111, 827, 431, 569, 521, 1607, 1151, 137,
1481, 1049, 881, 197, 1031, 11, 2087, 617,
269, 347, 1019, 1697, 1877, 59, 659, 1427

S=7354

n=9 (not minimal ?)

1319, 5, 107, 71, 3389, 2027, 2081, 3251, 29,
11, 2129, 1229, 521, 3119, 1289, 881, 311, 2789,
101, 17, 1049, 2999, 179, 3371, 2339, 2087, 137,
1787, 617, 431, 1931, 239, 197, 2729, 1019, 3329,
1277, 2381, 1481, 191, 2267, 1667, 1301, 1487, 227,
3167, 461, 3299, 1451, 281, 1061, 821, 1697, 41,
3257, 2711, 1877, 809, 659, 641, 347, 827, 1151,
1091, 3539, 857, 1619, 149, 1427, 59, 569, 2969,
269, 419, 1949, 2687, 1997, 599, 1721, 1031, 1607

S=12279

n=10 (minimal)

41, 2657, 2129, 149, 1997, 3539, 827, 2381, 1787, 1277,
1721, 3371, 3851, 179, 3359, 599, 1451, 137, 2087, 29,
1619, 1667, 3671, 71, 191, 2729, 2267, 2081, 1871, 617,
3461, 227, 1301, 461, 17, 1877, 3581, 11, 3299, 2549,
1151, 2801, 1487, 2111, 1031, 431, 101, 2969, 881, 3821,
197, 659, 59, 3119, 3257, 2339, 3557, 2789, 569, 239,
3389, 1481, 2309, 1949, 1319, 311, 1061, 1931, 1427, 1607,
419, 2999, 821, 2591, 857, 2711, 1697, 269, 1091, 3329,
3767, 281, 347, 2687, 1229, 2141, 5, 1049, 3251, 2027,
1019, 641, 809, 3467, 3527, 107, 2237, 3167, 521, 1289

S=16784

See
http://www.primepuzzles.net/puzzles/puzz_080.htm

The non-minimal pandiagonal 7×7 square of twin primes:

17, 5279, 7589, 37361, 3371, 44069087, 17189,
34031, 44066819, 6197, 13679, 4019, 1319, 13829,
17681, 59, 7559, 10499, 44097479, 3929, 2687,
44073947, 34589, 419, 6689, 13721, 6299, 4229,
2729, 19961, 2969, 44067677, 11057, 31079, 4421,
4787, 7547, 35081, 461, 8969, 16631, 44066417,
6701, 5639, 44080079, 3527, 1277, 11549, 31121

S=44139893.

See
http://www.primepuzzles.net/puzzles/puzz_689.htm

Is required for puzzle # 80:

1. find a solution for n = 7 with a magic constant S < 44139893;
2. find a solution for n = 9 with a magic constant S < 12279.

• Questa risposta è stata modificata 9 anni, 4 mesi fa da Natalia Makarova.

[Natalia Makarova]

I found a solution!

n = 7 (not minimal ?)

821, 281, 599, 347, 1451, 827, 1667, 
29, 659, 5, 227, 1949, 1427, 1697, 
881, 1721, 101, 1607, 311, 1301, 71, 
1787, 1061, 1229, 239, 857, 179, 641, 
1049, 521, 569, 1877, 1091, 269, 617, 
107, 461, 1871, 1277, 197, 1931, 149, 
1319, 1289, 1619, 419, 137, 59, 1151

S = 5993

My algorithm

Here you can see an scheme of the magic square of order 7:

ai (i = 1, 2, 3, …, 34) – independent variables, xk (k = 1, 2, 3, …, 15) – dependent variables.
Magic constant S is given.

The general formula of the magic square of order 7:

x1=S-a1-a2-a3-a4-a5-a6
x2=S-a7-a8-a9-a10-a11-a12
x3=S-a6-a12-a16-a18-a21-a26
x4=S-a1-a8-a14-a18-a22-a30
x5=S-x3-a31-a32-a33-a34-x4
x6=S-a25-a26-a27-a28-a29-a30
x7=S-a6-x2-a17-a24-x6-x4
x8=S-a4-a10-a15-a18-a28-a32
x9=S-a5-a11-a16-a22-a29-a33
x10=S-a3-a9-a14-a21-a27-a31
x11=S-x1-a12-a19-a23-a30-a34
x12=S-a13-a14-a15-a16-x11-a17
x13=S-a1-a7-x12-a20-a25-x3
x14=S-x13-x10-a18-x9-a19-x7
x15=S-a2-a8-a13-x14-a26-x5

For details see the article:
http://www.natalimak1.narod.ru/formul2.htm

Now we need to find a solution for n = 7 with a magic constant S < 5993.

See also
http://dxdy.ru/post942305.html#p942305

[Natalia Makarova]

I found a minimal solution for n = 7.

419, 1061, 881, 71, 569, 107, 1301,
17, 641, 821, 179, 1031, 1289, 431,
1427, 41, 269, 1151, 191, 521, 809,
1229, 857, 461, 659, 827, 137, 239,
599, 1607, 347, 1319, 281, 29, 227,
101, 5, 1481, 11, 1451, 1049, 311,
617, 197, 149, 1019, 59, 1277, 1091

S=4409

Now we need to find a solution for n = 9 with a magic constant S < 12279.
There is a solution?

[Natalia Makarova]

Progress!

1619, 1487, 179, 2027, 617, 827, 1949, 2657, 11,
1151, 2549, 191, 1061, 2687, 599, 1697, 1289, 149,
821, 269, 239, 2339, 857, 29, 1319, 2789, 2711,
1931, 1277, 2801, 1091, 641, 2111, 1019, 461, 41,
2267, 1427, 809, 431, 1301, 1871, 659, 521, 2087,
1787, 1229, 1877, 2309, 311, 2081, 17, 281, 1481,
1031, 59, 2729, 101, 227, 1997, 881, 3299, 1049,
569, 107, 2129, 1667, 2141, 137, 2381, 5, 2237,
197, 2969, 419, 347, 2591, 1721, 1451, 71, 1607

S = 11373

I guess it is a minimal solution.

[Natalia Makarova]

Published a new puzzle:

Magic squares and consecutive twin primes

http://www.primepuzzles.net/puzzles/puzz_769.htm

I invite all to solve this problem for n = 3, 4, 7.

[Natalia Makarova]

My colleague Serg Zorkin found minimal solution for n = 7

431  2267 2237 347  1487 2087 419
2129 2027 569  461  1427 2141 521
857  827  1931 1721 881  1607 1451
1289 821  809  1949 1871 659  1877
1091 1667 1787 1019 1061 1031 1619
1997 617  641  2081 1229 599  2111
1481 1049 1301 1697 1319 1151 1277

S=9275

[Autore]

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