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A Curious Property of Prime Numbers.
(By T. S. Barrett, London, Eng.)
There is a remarkable property of all prime numbers (excepting i and 2) not possessed by other numbers. To explain, it will need a few introductory words.
Write down, as in the top row, contiguously, as many consecutive numbers as may be desired, commencing with unity. Repeat them in the next row, with one vacant cell between each; in the third row, with two vacant cells between each; and so on ad libitum. If we now imagine some object (a chess-man, for example) to start from the corner cell at the top marked with a star, and to travel along any of the paths indicated by the figures, it will be seen that each path is distinct and well defined. If it travels downward along the cells occupied by 1, the path is a column. If it goes from 2 to 2, the route is a diagonal. If it jumps to a 3 in the top row, thence to the three in the next row, and so on, the path is a knight's move." Similarly with the other numbers. We may distinguish the different paths by calling them No. I path, No. 2 path, No. 3 path, and so on.
Now if we select any prime number, and strike out from the diagram all numbers greater, and divide the whole perpendicularly into sections, each containing as many columns as numbers chosen, the various sections may be superimposed without confusion,-every num
ber in the outside sections falling into an empty cell, excepting when the number of the row corresponds to the prime number selected, or any of its multiples. To make this clearer, let us take the number 7, and cancel all numbers greater. Then our first section will be up to and including the column with 7 at the top ; the next will be from the column, previously with 8 at the top, up to aud including the column that was headed by 14. And so on.
The above is the arrangement in the first section of all the outside numbers greater than 7. I have supposed that the diagram has been continued far enough to fill the above ; but the principle is just the same, Thus the 5 in the second row of the first diagram being in the second column of the second section, must be transferred to the corresponding cell of the first section. It falls into the vacant place between the i and the 2. The 6 in the same row falls for the same reason, into the vacant place between the 2 and 3. Similarly with the third row. The 4 being in the third column of the second section, has to be transferred to the corresponding cell, and falls into the vacant place just before the 2 in the third row. Similarly with the 5. The 6 in the same row is in the second column of the third section, and therefore falls into the vacant cell between the i and the 4. In the same way with all the other rows, until we come to the seventh, which corresponds with the chosen prime number. Then the num. bers in that row instead of falling into vacant places, all crowd into the cell we have indicated by a dagger. After that the rows are repetitions of what has been given above; the first and eighth being alike, and so on.
Many curious things may be observed in the above diagram ; and all others made similarly from other prime numbers. If we obliterate the column with 1 in it, and also the dagger row and all beneath it, we have remaining a very curious square ; these are among its other properties : all numbers equally distant fiom the center, both in rows and columns, add the same, namely, 9 (or the prime number plus 2). Every column and every row contains all the numbers from 2 to 7; and the diagonals consist only of 2 and 7. Now, if we have any square consisting of cells, the root being a prime number, we may deduce from the foregoing facts one having a bearing on magic squares with odd roots. If we cancel the last row of the second diagram, as well as all the numbers in it, we get an empty square renaining, which we may proceed to refill with the figures in another way. proceed from the cell marked with a dagger and completely fill the square with figures along different paths, without any collision. Thus, we may fill the lowest row by proceeding in a horizontal direction. Let us fill it with ciphers and call it the " zero path.”
We may proceed in an upward direction and fill the first column. This will be the No. I path. Diagonally we may proceed along the cells occupied by the figure 7. This is No. 2 path. Then we may fill in along the No. 3 (on the “ knight's ") path. We may put a 3 (for example) along this path, and it will occupy the cells where the 6 was; and so on. The root of the square being 7, we may thus fill the square with figures along 8 different roots, always i more than the root of the square. It must be observed that when a path runs off at the right side of the square, before the top row has been reached, the figure must be brought back within the square, in a similar manner to that already explained when superimposing the sections of the first diagram.
Thus, take the figure 6 in the first diagram, the figure at the end of the fourth row. The path taken by that number being No. 3, it would bring us, if there were more cells to the right, into the second cell of the fourth row.
Hence the next cell to be filled must be the corresponding cell inside the square.
Now, many writers on magic squares, especially M. de La Hire, must have had an idea of these properties ; but the credit of the discovery is generally conceded to the Rev. A. H. Frost of Nasik, in India. He pointed out that M. de La Hire's method of making oddroot magic squares necessarily follows from these properties of the "paths." A.
Thus, in the square A, all the numbers are in No. 3, or the" knight's" path. Take the 4 in the bottom row, and thence to 4 in the row above, and so on. The 2 and 3 run off at the side, and are brought back within the square. Now the square B, on the other hand, has its numbers connected by the “ No 4. path.” Notice from 40 to 40 for instance. Consequently the differences between the numbers in square B, being not less than 5, a magic square is necessarily produced. For we have already seen that where prime numbers are concerned, the different “paths never clash except in one spot. Therefore, one square being in “ No. 3," and the other in “No. 4 path,” any two numbers once united will never meet again in the same square. Thus, for example, the 3 in the one square and the 20 in the other meet on superposition once and once only. If they met oftener, or not at all, the combined square would not be magic. There is another property of these “nasik” magic squares, whose roots are prime numbers.
The magic summation may be counted along any path," excepting those two paths chosen for the primary squares. The square of 5 can only have 6 paths, namely, the horizontal row, the perpendicular column, two diagonal or slanting rows in opposite directions, No. 3, and No. 4 paths. But as the two latter have to be chosen in a root-5 square, for the primary squares, the magic square of 5 cannot have its summation in any route excepting rows, columns, and diagonals. But in the square of 7 below, which may have two more paths than the 5 square, summation may be made along two additional ways.
(By the bye, there is a curious feature in this square not often met with, following from the way I have made it. If every number has its figures (e. g. 14 changed into 41, 32 into 23, and so on), the square remains magic.)
The above magic square not only sums 308 in the usual ways, but if the numbers according to No. 3 path (or “knight's move ") be taken, likewise those along No. 4 path, they will likewise sum 308. But care must be taken that in making these moves that no row is