Multimagic series for squares
See also the multimagic series for cubes

As we have seen for the smallest bimagic and the smallest trimagic squares, it may be interesting, in order to try to construct a p-multimagic square of order-n, to find all the p-multimagic series of order n, that is to say the series of n different integers, from 1 to n, having the correct magic, bimagic,... up to p-multimagic sums.

The order 4 is the smallest order allowing us to get a bimagic series. Here are the 2 bimagic series, which are also surprisingly trimagic:

This means that:

  1. 15 + 9 + 8 + 2 = 14 + 12 + 5 + 3 = 34 = S1
  2. 152 + 92 + 82 + 22 = 142 + 122 + 52 + 32 = 374 = S2
  3. 153 + 93 + 83 + 23 = 143 + 123 + 53 + 33 = 4624 = S3

For the order 5, there are 8 bimagic series for which the list is given in the Smallest bimagic square page.

Here is a summary of the number of multimagic series. Some of these lists are downloadable as Excel files, from 28Kb to 800Kb each. For order 12, the numbers of the trimagic, tetramagic and pentamagic series were kindly communicated by Walter Trump, Germany. In 2004, the numbers of Walter Trump were confirmed by Fredrik Jansson, Finland. And Fredrik goes further: he is the first to have computed the huge number of bimagic series of order 12, and the number of multimagic series of order 13 (excepted bimagic series of order 13).

New steps on bimagic series, between July and October 2005, from Germany:

In April 2008, Michael Quist, USA, computed the numbers of tetramagic and pentamagic series of order 16. In May 2008, he computed the number of trimagic series of order 15 (and also confirmed the number of trimagic series of order 13 previously computed by Fredrik Jansson). In August 2008, he computed the number of bimagic series of orders 18, 19 and 20 (and also confirmed the number of bimagic series of smaller orders previously computed). In May 2013, he computed the number of bimagic series of orders 21, 22, 23, numbers independently computed and confirmed a few days later by Lee Morgenstern, USA. Then in May and June 2013, directly using the software written by Lee Morgenstern (see his counting method), Walter Trump computed the number of bimagic series of orders 24, 25, 26, 27 and 28. In December 2014, Lee Morgenstern computed the number of trimagic series of orders 16 and 17. In August 2015, Dirk Kinnaes, Belgium, confirmed the previously known numbers of bimagic series, and computed the numbers of bimagic series of orders 29 and 30: look at his algorithm (PDF file).

The order 12 is the smallest order allowing tetra and pentamagic series. In the 106 tetramagic series, 4 of them are also pentamagic. These 4 series are symmetrical left/right, that is to say that i-th number + (13-i)th number = 12 + 1 = 145.

What is the smallest order allowing hexamagic series? The smallest current candidates are orders 27 and 40.

The bimagic, trimagic, tetramagic and pentamagic series are referenced respectively under the numbers A052457, A052458, A090037 and A106646 in the On-Line Encyclopedia of Integer Sequences, OEIS Foundation.

See also the "Multimagic Series" topic in the Eric Weisstein's World of Mathematics, Wolfram Research.

In April-May 2005, Walter Trump computed an estimation of the number of various magic series, including bimagic series from order 13 to order 20, using Monte Carlo methods. More details at The true numbers of bimagic series, computed after his estimation (see above), are very close: his estimation method has proved excellent!

In June 2013, Michael Quist wrote a paper estimating the numbers of magic and multimagic series, including bimagic series for squares of order N. See Here is his formula, and the numeric values obtained for 20 ≤ N ≤ 30. The error decreases with higher orders. In August 2015, Dirk Kinnaes improved Quist's formula, adding a new term (see also his PDF given above).

Not in his paper, but in May 2013 Michael Quist calculated this other formula:

The numbers estimated by this formula, for N > 18, are in the trimagic column of the first table of this page.
And in December 2013, he proposed this general formula for multimagic series of order N (the trimagic formula above can be obtained with K=3):

In January 2006, Robert Gerbicz, Hungary, proved that there is no tetramagic series of order 15. Here is his proof.

The magic sums are:

S4==15 mod 16. Because (2x+1)^4==1 mod 16 and (2x)^4==0 mod 16, all 15 numbers have to be odd.

Let a(k)=2b(k)+1 (where every b(k) is an integer) and T1=sum(b(k), k=1..15), T2=sum(b(k)^2, k=1..15), and so on...

From the binomial theorem we have:

It is easy to solve this linear system of equations:

There is a contradiction. Because x==x^4 mod 2, we should have T1==T4 mod 2: but T1 is even and T4 is odd!

So there is no tetramagic (and pentamagic, hexamagic...) series for squares of order 15.

In July 2008, Jaroslaw Wroblewski, Poland, proved that there is no hexamagic series of order 17. Here is his proof.

From S4=1(mod 17) we conclude that the series must contain 17 odd terms or one odd term and 16 even terms.

***** In case of 17 odd terms, each of the form 4k+1 (where k may be negative)(*), after denoting by Ki the sum of i-th powers of k's, we get

which leads to

giving a contradiction on parity of K1. There are no tetramagic series in this case.

***** In case of 1 odd term of the form 4k+1 (where k may be negative) and 16 even terms of the form 2p, after denoting by Pi the sum of i-th powers of p's, we get from equations on S2 and S4 respectively:

As P2 and P4 have the same parity, k must be even, say k = 2m. We get from equation on S6:

which allows us to write m=4r. We get from equation on S2 and S6:

Since P2=P6(mod 4), we can take r=4q. At this point the odd term of the magic series is equal to 128q+1, q = -2,-1,0,1,2.

From the equation on S4, which takes form

we conclude that P4 is divisible by 32 and in view of P4 = P6 (mod 8), P6 is divisible by 8.

Equation on S6 gives:

This forces q to be odd, i.e. q must be plus or minus 1.

In case q=1 we get:

In case q=-1 we get:

In either case P4 is divisible by 16, which means that all p's are odd or all are even. However P6 is not divisible by 64, which rules out the possibility of all even p's. Let p=4x+1 (x being integer of any sign).

From equations on S2 and S4 we get for q=1:

and for q=-1:

In both cases there is mod 4 contradiction.

And again in July 2008, Jaroslaw Wroblewski proved that there is no hexamagic series of order 19. Here is his proof.

S4(mod 16)=7, implies that each series must have 7 odd and 12 even terms, of the form 4k+1 and 2p respectively. Then equations on S2, S4, S6 give:

The last equation gives even K1 and then P2 and P4 get contradictory parity, which proves there are no hexamagic series of order 19.

(*) About odd terms of the form 4k+1 "WHERE k MAY BE NEGATIVE". Yes, each odd integer, or its opposite, is of the form 4k+1:

and so on, with k = 0, -1, 1, -2, 2, -3,... Nice trick!

From March to August 2013, Lee Morgenstern proved that there is no tetramagic series of orders 3, 4, 5, 7, 8, 9, 11, 15 and 17. Proved that there is no pentamagic series of orders 19, 23 and 25. And proved that there is no hexamagic series of orders 12, 13, 16, 19, 20, 21, 23, 24, 25, 28, 29, 31, 32, 33, 35, 36, 37 and 39.

The existence of hexamagic series of orders 27 and 40 (and numerous >40) is unknown.

In July and August 2013, he proved that some multimagic series exist, finding these examples:

In October 2013, Christian Boyer found this symmetrical family:

and from Lee Morgenstern:

In September-October 2013, Jean Moreau de Saint-Martin, France, studied and checked in detail Morgenstern's mathematical proofs on multimagic series (from the above zipped text files), and found no error.

And he went further, simplifying and generalizing Morgenstern's proofs. Also using some of my remarks and computations, we now have new impossibility proofs:

 Here is this study on multimagic series:

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