Magic sequences


This page is dedicated to magic sequences, to form regular magic squares. ie generalized writable as:
nIN-Magic Squares {f(i) = x0 +/- aid (i = 1..n')|'sums' = magic number}
relation between n' and n not quite clear yet (n' = n+1 (for n==3)),
'sums' are the regular magic sums

In response to the puzzle Prime magic square puzzle #3
I analysed the there uploaded Nelson Sequence as being of the following generalised form:

f(i) = x0 +/- aid with x0 {ai}d = 1480028171 {2,3,5,7}6 (introducing the here used notation)
The following table give the neccessairy conditions for forming an order 3 square

a1 + a2 = a3
a1 + a3 = a4
a1
a2
-a3
-a4
0
a4
a3
-a2
-a1

notice that the above conditions are not only satisfied by the nelson sequence but also by the numbers used in a regular order 3 square (ie 5 {1,2,3,4}1) or any number sequence in arithmetic progression. In all cases the magic number is thrice the central number so ths gives:

3IN-Magic Squares {f(i) = x0 {a1,a2,a3,a4}d (a1 + a2 = a3;a1 + a3 = a4)|'sums' = 3 x0}

notice also that the above matrix is obtained from a regular order 3 square, subtracting 5 from all numbers, this gives both the sign and the index corresponding to the square above Harvey Heinz: Prime-Magic Stars. uploaded the above mentioned 1480028171 {2,3,5,7}6 and a simular 185050099 {2,3,5,7}6
he also told me the other nelson sequences which have some other ai sequence fully tabulized as:

primesquares by Nelson
1480028129 + 42
1850590057 + 42
5196185947 + 42
5601567187 + 42
5757284497 + 42
6048371029 + 42
6151077269 + 42
9517122259 + 42
19052235847 + 42
20477868319 + 42
24026890159 + 42
28519991387 + 42
34821326119 + 42
44420969909 + 42
73827799009 + 42
73974781889 + 42
76483907837 + 42
76560277009 + 42
85892025227 + 42
95515449037 + 42
{2,3,5,7}6
23813359613 + 84
{5,4,9,14}6
49285771679 + 72
{5,2,7,12}6

The given number above is the start of a consecutive series of 9 primenumbers
The added number is to devert from the start of the primenumber sequence to the central number of the square. the numbers I pasted in from the email Heinz sent me, and remain currently unverified since I have no way to verify the primality of the numbers above, also the consecutiveness of the sequences (no primes in between) I have no means to verify. The magicness of the square is automatic by the above mentioned theory.


Harry J. Smith defined type 1 and type 2 order 3 magic squares. The regular order 3 square is defined as being type 1. In the notation used above type 1 and type 2 can be defined according to the following table (which also lists the smallest example)

Order 3 types magic squares
Type 1
a1 < a2
5 {1,2,3,4}1
Type 2
a1 > a2
6 {2,1,3,5}1
6
7
2
8
7
3
1
5
9
1
6
11
8
3
4
9
5
4

According to this distinction the last two prime number magic square are of type 2 while all the others are of type 1.

according to Magic square of 3 x 3 prime number
the following 92 decimal number:
9967943206670108648449065369585356163898236408
0991618395774048585529071475461114799677694651
+ 1020 is the base of an order 3 prime number magic square with sequence {1,2,3,4}210 found by Mr. Dennis Kluk.


subtracting the two basic relations:
a1 + a2 = a3
a1 + a3 = a4
gives after some little rearangement:
a2 + a4 = 2 a3
interestingly this relation between a cornerpoint and its knightjump conected numbers is independent of the central number x0 and a needed relation between the numbers in any order 3 magic square.


According to Chebrakov's result mentioned in:
The Prime Puzzles and Problems Connection puzzle #79
a magic square consisting of square numbers:
3IN-Magic Squares {f(i)=mi2 (mi in IN)}
can't exists

My analysis into the matter let me to define
Pseudo Magic Rings
I have not (yet?) found a definite proof of the above theorema but I narrowed it down to proving that a ring of square numbers can't be a 3R0 Rings of square numbers do exist (see the above file)