Prime Magic Stars

Welcome to magic stars formed with prime numbers

The Minimal Solution Order-5 Prime Magic Star

It is possible to construct a magic order-5 star using prime numbers. The minimal solution uses the following 10 prime numbers : 5, 7, 11, 13, 17, 19, 23, 31, 41, 43. Note that the primes 3, 29 and 37 are missing from this series. The only even prime number, 2, of course cannot be included because of parity.

The sum of the above prime series is 210 and the magic sum is 84 (S = 210 * 2/5).

The smaller series of primes from 3 to 41, again leaving out the 29 and 37, is a candidate because the sum is also evenly divisible by 5. However, there are not enough suitable combinations of 4 numbers summing to the constant 68 to permit construction of a magic star.

There are 12 basic solutions using the above set of prime numbers. Because each solution has 4 rotations and 5 reflections, there are 120 apparently different solutions. These basic solutions were first reported by Vasiliy Danilov on August 3, 1998 to Carlos Rivera in response to a puzzle posted on his Prime Puzzles web page at http://www.primepuzzles.net.

The 12 basic solutions are (in index order): Primes used  are 5, 7, 11, 13, 17, 19, 23, 31, 41, 43

 #    A   b   c   D    e   f   G    h   I   J  sum  Pcomp
 1 .  5   7  31  41   13  19  11   43  23  17  84    9
 2 .  5   7  41  31   13  17  23   43  11  19  84   10
 3 .  5  17  19  43   11   7  23   31  13  41  84    8
 4 .  5  17  43  19   11  41  13   31  23   7  84   11
 5 .  5  19  17  43   23   7  11   41  13  31  84    7	reported by Rivera &
 6 .  5  19  43  17   23  31  13   41  11   7  84   12        Ayala
 7 .  7   5  31  41   13  11  19   43  17  23  84    5
 8 .  7   5  41  31   13  23  17   43  19  11  84    3
 9 .  7  11  23  43   17   5  19   41  13  31  84    1
10 .  7  23  11  43   19   5  17   31  13  41  84    2	reported by Vasiliy
11 . 11  13  19  41    5   7  31   23  17  43  84    4
12 . 19  13  11  41    7   5  31   17  23  43  84    6

Pcomp

As noted above, the basic solutions come in Pcomp (peak complement?) pairs, which can be transformed from one to the other by the following exchange routine. This moves the numbers at the peaks to the valleys, and the numbers in the valleys to the peaks.

Position A and E change places.
B moves to I,   C moves to J,    d moves to C,   F moves to D
G moves to B,   h moves to G,    I moves to H,   J moves to F

After these changes have been made, an additional step is required to convert from an equivalent solution to the new basic solution. This is called normalizing, and involves rotating one or more positions, and possibly reflecting the pattern as well, so the above conditions for a basic solution are met. I call these pairs Pcomp to differentiate from complement as it is normally used when dealing with pure magic squares and pure magic stars.

Vasiliy has a different routine for accomplishing this, but after normalizing, the end result is the same basic solution pair. Note that these pair members may be called compliments of each other. However, this is different to the compliment referred to in normal magic stars (or magic squares) where the compliment is obtained by subtracting each number from the sum of the first and last primes in the series used.. The result is sometimes, but not always, the same as using an exchange procedure like the above.

Note that like all magic squares and magic stars, prime magic stars may also be complemented (the conventional way) to produce another magic star. However, the complement of many of the prime numbers will not be a prime, so the result is only an impure magic star.

Other near minimum solutions

Prime series used: 3, 5, 7, 11, 13, 29, 31, 41, 47, 53          (primes 3 to 53 with no 17,19,23,37)

Prime series used: 7, 11, 13, 17, 19, 23, 29, 31, 43, 47             (primes 7 to 47 with no 3, 5, or 37)

Note that one of these three series of 12 solutions starts with the prime number 3, one with prime 5, and one with prime 7.
There is another set of 12 solutions using primes 5, 7, 13, 19, 23, 29, 37, 41, 43, 53 with S = 108.
There is another set of 12 solutions using primes 5, 11, 13, 29, 31, 37, 41, 43, 47, 53 with S = 124.
There is another set of 12 solutions using primes 5, 13,17,  23, 31, 37, 41, 43, 47, 53 with S = 124.
Therefore there are a total of 72 distinct solutions using 10 of the 15 primes from 3 to 53.

The lowest possible Magic Pentagram using consecutive primes

There are also 12 basic solutions to the order-5 consecutive primes magic star. This time the first set that works starts with the 1644th prime. The series of 10 consecutive primes start with 13907 and ends with 14009. It is worth noting that there are also 12 basic solutions to the minimal solution Prime Number Order-5 magic stars (above section). There are also 12 basic solutions to the minimal solution Order-5 magic stars. These use numbers 1 to 12 with no 7 or 11, sum = 24 and using numbers 1 to 12 with no 2 or 6, sum = 28. There are no solutions using the consecutive numbers 1 to 10.

 #    A     b     c     D     e     f     G     h     I     J     Sum   Pcomp
 1  13907 13913 13997 13999 13921 13933 13963 14009 13931 13967  55816	10
 2  13907 13913 13999 13997 13921 13967 13931 14009 13963 13933  55816	9
 3  13907 13933 13967 14009 13931 13913 13963 13999 13921 13997  55816	7
 4  13907 13933 14009 13967 13931 13997 13921 13999 13963 13913  55816	11
 5  13907 13967 13933 14009 13963 13913 13931 13997 13921 13999  55816	8
 6  13907 13967 14009 13933 13963 13999 13921 13997 13931 13913  55816	12
 7  13913 13907 13997 13999 13921 13963 13933 14009 13967 13931  55816	3
 8  13913 13907 13999 13997 13921 13931 13967 14009 13933 13963  55816	5
 9  13913 13931 13963 14009 13933 13907 13967 13997 13921 13999  55816	2
10  13913 13963 13931 14009 13967 13907 13933 13999 13921 13997  55816	1
11  13931 13921 13967 13997 13907 13913 13999 13963 13933 14009  55816	4
12  13963 13921 13933 13999 13907 13913 13997 13931 13967 14009  55816	6

Carlos Rivera and Jaime Ayala's solution to this part of their puzzle was:
a = 13907, b = 13921, c = 13913, d = 13933, e = 13931, f = 13963, w = 13999, x = 13997, y = 13967,
z = 14009. When changed to my notation and normalized, it is the same as my index # 5.

Sample solutions from other consecutive prime sets

1 of 12 solutions using 10 consecutive primes starting at 23497
23497 23537 23563 23567 23509 23549 23539 23557 23531 23561 94164

1 of 12 solutions using 10 consecutive primes starting at 23831
23831 23869 23893 23899 23833 23873 23887 23879 23857 23909 95492

1 of 12 solutions using 10 consecutive primes starting at 29423
29423 29429 29501 29527 29443 29473 29437 29531 29483 29453 117880

There are 12 solutions for each series starting with 37447, 39313, 40813, 81331, 105361, 112207, 115303, 122453, etc.

Minimal solution Order-6 prime magic star using non-consecutive primes

Primes used: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47        These are all the basic solutions.

#   A   b   c   D   e   f   G   h   i   J   K   L   sum  type
1.. 3   7  31  41   5  23  13  19  47  17  29  11   82	 1a  see notes (below)
2.. 3   7  31  41  17  11  13  19  47   5  29  23   82	 2a   re Suzuki types
3.. 3  13  19  47  11  17   7  31  41  23  29   5   82	 1b
4.. 3  13  19  47  23   5   7  31  41  11  29  17   82	 2b
5.. 5  23  13  41   3   7  31  17  29  19  47  11   82	 1c
6.. 5  23  47   7   3  41  31  17  29  19  13  11   82	 2d  too bad 2c & 2d 
7.. 7   3  41  31   5  29  17  11  47  13  23  19   82   1d   are not reversed!
8..17  11  13  41   3   7  31   5  29  19  47  23   82	 2c

None of these 8 solutions can be converted so peaks and valleys are swapped (i.e. no pcomp pairs).
Remember, when converting Suzuki types, you must rotate and/or reflect the pattern to obtain the new basic solution.

Other almost minimal solutions

Primes used: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 53     These are all the basic solutions

1 .. 3   5  23  53  13   7  11  41  29  31  17  19   84	 1a
2 .. 3  11  41  29  19  31   5  23  53   7  17  13   84	 1b
3 .. 5   3  53  23  13  17  31  19  29  11   7  41   84	 1c
4 ..13   7  11  53   3   5  23  31  17  41  29  19   84	 1d

Primes used: 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 47, 59    These are all the basic solutions

1 .. 3   5  23  59  13   7  11  47  29  37  17  19   90  1a
2 .. 3  11  47  29  19  37   5  23  59   7  17  13   90  1b
3 .. 5   3  59  23  13  17  37  19  29  11   7  47   90  1d
4 .. 3   7  11  59   3   5  23  37  17  47  29  19   90  1c

Primes used: 3, 5, 7, 11, 13, 17, 19, 23, 29, 41, 43, 59    These are all the basic solutions

1 .. 3   5  59  23   7  19  41  17  29  13  11  43   90  1a
2 .. 3  23  59   5  13  43  29  17  41   7  11  19   90  1b
3 .. 5   3  23  59   7  11  13  43  29  41  19  17   90  1d
4 .. 7  11  13  59   5   3  23  41  19  43  29  17   90  1c

NOTES

Type refers to Suzuki groups. The number is the index number of A. B, C & D are transformations of A.
See my Magic Stars Order-6 page for a full explanation of Suzuki groups and Complement pairs.
It appears there are always a multiple of 4 basic solutions for order-6 magic stars, because all may bedivided into Suzuki sets.

Pcomp refers to the opposite member of the pair which has the peaks and valleys reversed. I use the term Pcomp to differentiate from the term complement as it is normally used when dealing with pure magic squares and pure magic stars.

Not all solutions of order-6 prime number magic stars are members of pairs (unlike all pure magic stars). This is probably because any magic hexagon requires that the set of 12  numbers be capable of dividing into 2 sets of six numbers with the same sums,  and 3 sets of four numbers with the same sums.   Furthermore, in the case of prime numbers (or other non-consecutive number sets),each of the sets of 6 must be formed of  2 sets of three numbers with the same sum. Most of the prime sets cannot be arranged this way. Of those that can, most have not the correct values to permit six lines with the same sum.

In the case of the above low prime number series, there are no Pcomp pairs.

Solutions for Order-6 using consecutive primes 29-73

This is the smallest series of consecutive primes that can make an order-6 magic star. Primes used: 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71 & 73    There are 20 basic solutions. Most of these solutions do not come in pcomp pairs, unlike all pure magic stars, and all the solutions for the order-5 prime number magic star.

 #   A   b   c   D   e   f   G   h   i   J   K   L    sum   type   Pcomp
 1.. 29  31  71  73  43  41  47  67  61  53  37  59   204   1a
 2.. 29  37  71  67  31  47  59  43  73  41  61  53   204   2a
 3.. 29  37  71  67  41  53  43  73  59  61  31  47   204   3a	   16
 4.. 29  37  71  67  43  53  41  73  61  59  31  47   204   4a     15
 5.. 29  41  73  61  47  59  37  71  67  53  31  43   204   4b	   17
 6.. 29  43  73  59  47  61  37  71  67  53  31  41   204   3b	   19
 7.. 29  47  61  67  41  37  59  43  73  31  71  53   204   5a
 8.. 29  47  67  61  59  53  31  71  73  41  37  43   204   1b
 9.. 29  59  43  73  53  31  47  61  67  37  71  41   204   5b
10.. 29  59  43  73  53  41  37  71  67  47  61  31   204   2b
11.. 31  29  73  71  43  37  53  59  61  47  41  67   204   1d
12.. 31  47  59  67  29  37  71  41  61  43  73  53   204   2c
13.. 31  53  73  47  29  67  61  41  71  43  59  37   204   5d
14.. 37  29  67  71  31  61  41  53  73  59  47  43   204   2d
15.. 37  29  67  71  41  31  61  47  59  43  53  73   204   3d	    4
16.. 37  29  67  71  43  31  59  47  61  41  53  73   204   4d	    3
17.. 41  31  61  71  37  29  67  43  53  47  59  73   204   3c	    5
18.. 41  37  59  67  29  47  61  31  71  43  73  53   204   5c
19.. 43  31  59  71  37  29  67  41  53  47  61  73   204   4c 	    6
20.. 43  37  53  71  31  29  73  47  41  59  61  67   204   1c

The following solutions use consecutive primes 53 to 103
 #  A   b    c    D   e   f    G    h    i   J   K    L    sum   type  Pcomp
 1..53  61   97  101  67  71   73  103   83  89  59   79   312   1a    6
 2..53  61   97  101  73  71   67  103   89  83  59   79   312   2a    5
 3..53  67  103   89  79  83   61   97  101  71  59   73   312   2b    7
 4..53  73  103   83  79  89   61   97  101  71  59   67   312   1b    8
 5..61  53  101   97  67  59   89   79   83  73  71  103   312   1d    2
 6..61  53  101   97  73  59   83   79   89  67  71  103   312   2d    1
 7..67  59   89   97  61  53  101   73   71  79  83  103   312   1c    3
 8..73  59   83   97  61  53  101   67   71  79  89  103   312   2c    4
This is the only set for order-6 that I have found where all the solutions are 
in pairs. See above section for my definition for Pcomp.

The following solutions use consecutive primes from 59 to 107
 #   A   b   c    D   e    f    G    h    i    J    K    L    sum   type  Pcomp
 1.. 59  61  103  107  67   83   73   97  101   89   71   79   330   1a    
 2..59  61  103  107  71   79   73  101   97   89   67   83   330   2a
 3..59  67   97  107  61   73   89  103   79  101   71   83   330   3a
 4..59  67   97  107  61   83   79  103   89  101   71   73   330   4a
 5..59  73   97  101  79   89   61  103  107   83   71   67   330   1b
 6..59  73  101   97  83   89   61  103  107   79   67   71   330   2b
 7..59  79  103   89  73  101   67   97  107   83   71   61   330   4b
 8..59  89  103   79  83  101   67   97  107   73   71   61   330   3b    16
 9..61  59  107  103  67   71   89   79  101   73   83   97   330   1d
10..61  59  107  103  71   67   89   83   97   73   79  101   330   2d
11..61  71  101   97  67   59  107   79   83   73   89  103   330   4c
12..61  71  101   97  67   59  107   89   73   83   79  103   330   3c
13..67  59  107   97  61   71  101   73   89   79   83  103   330   4d
14..67  59  107   97  61   71  101   83   79   89   73  103   330   3d
15..67  71   89  103  61   59  107   73   83   79  101   97   330   1c
16..71  67   89  103  61   59  107   73   79   83   97  101   330   2c     8

Others solution sets start with 127 (8), 137 (16), 409 (12), 541 (12) 

A Standard is Needed

Introduction

In early summer of 1998, Carlos Rivera posted a two part puzzle on his Prime Puzzles & Problems web page (at http://www.primepuzzles.net)

1. Find the minimal sum prime number magic pentagon using 10 not necessarily consecutive prime numbers.
2. Find a prime number magic star using the smallest possible set of 10 consecutive prime numbers.

When Carlos posted the puzzle he and his friend Jaime Ayala already had 10 solutions for part A and 1 solution for part B of the puzzle.

On August 3/98 Vasiliy Danilov sent Carlos six solutions for part A. of the puzzle, and explained how they could be transformed to 12 solutions.

Subsequently, I also worked on this puzzle.
The end result was three sets of solutions with three different notations. Here I compare them and make some suggestions for a standard.

Puzzle part A -- Minimal Sum Prime Number Magic Star
Example a.
Vasiliy Danilov:
a = 43, b = 13, c = 11, d = 19, e = 41, f = 23, g = 5, h = 31, i = 7, j = 17
Harvey Heinz (solution # 10)
a = 7, b = 23, c = 11, d = 43, e = 19, f = 5, g = 17, h = 31, i = 13, j = 41

These solutions are equivalent, but obviously they do not look like it, and it is impossible to compare them without actually drawing the star and assigning the values to the correct cells, as per the left and middle diagrams above.

When this is done, they are still not identical, but are only apparently different. One may be called a basic solution, the other an equivalent solution. Equivalent solutions are rotations and/or reflections of the basic solution. As there are 2 reflections and 5 rotations for the order-5 magic star, there are 10 apparently different solutions, of which only one is called the basic solution. In general, for any order n magic star, there are 2n-1 apparently different solutions for each basic solution.

The question now is, which solution do we call the basic solution?

I propose the following:
To qualify as a basic solution, two conditions are required:

1. The lowest value point number must be at the top (position A).
2. The top right hand valley must be lower in value then the top left valley.

We now see that by this definition, mine is the basic solution and Vasiliy's solution is an equivalent one. It must be rotated right (clockwise) two positions to become the basic solution. This process is referred to as normalization.

Example b.
Carlos Rivera & Jaime Ayala (1 of 10 solutions found)
a = 5, b = 11, c = 17, d = 7, e = 13, f = 23, w = 43, x = 19, y = 41, z = 31
Harvey Heinz (solution # 5)
a = 5, b = 19, c = 17, d = 43, e = 23, f = 7, g = 11, h = 41, i = 13, j = 31

For this example, assign the values to the stars as per the two layouts (middle and right, above). By the above definition, my solution is the basic one, Carlos & Jaime's is an equivalent solution. Change it to the basic solution by rotating right (clockwise) 3 positions. In this case the normalization is complete only after a reflection of the pattern is made.

We now have a definition for what constitutes a basic solution, except no mention was made regarding the order of recording the values.

I propose the values be assigned and recorded by tracing out the lines of the star as per the middle diagram (Heinz) above. Quite often you see these values presented horizontal line by line (Danilov). This is acceptable for Order-5 or 6 magic stars but becomes hopelessly confusing when higher order stars are considered.

Having a standard definition for what is a basic solution makes it possible for different investigators to compare their lists of basic solutions for a given order of magic star.

1.  All solutions have a unique index number.
2.  After normalization, it is immediately apparent if two solutions are equivalent because, after sorting,
         they will be identical and appear adjacent in the list.
3.  It is not necessary for comparison purposes,  to actually draw the diagram and assign the values to it.

Please refer to my Magic Stars Definitions page for  more detail.
Please refer to my Magic Stars -2 page for similar information on order 7 and 8 prime magic stars.

Please send me Feedback about my Web site!

Harvey Heinz   harveyheinz@shaw.ca
Last updated March 01, 2005
Copyright © 1998, 2000, 2001 by Harvey D. Heinz