Number Theory

Number Theory Madras 1987 by Krishnaswami Alladi

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By Krishnaswami Alladi

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2 It is interesting to note that God took six days to create the world, and then rested on the seventh. " The command was specific. Incorporating the number 7 at the appropriate place in a ritual was supposed to cause some of its power to be passed into the control of the practitioner. The most frequent method of incorporating numbers into magic ritual was by repetition of various parts of the ritual. For example, the following is a spell taken from /I 42 MATHEMATICAL MYSTERIES a modem book on magic which is supposed to bring back a lover who has been unfaithful: Who turns from you shall yet be bound If signs of him may still be found Within your house-one hair or thread, Fragment of color, scent or word, Or any thing that bears his touchThis spell turns little into much: Seal the relic in a box With seven strings tied round for locks, Each one tight knotted seven times: Then set on it these seven signs: Hide it in darkness, out of sight, Until the next moon's seventh night, Then send it to the one you seekHe must return within a weee Other numbers frequently used in magic ritual included 3, 5, and 9.

Hence, to calculate the square of 5 or 52 we add all odd numbers up to 2·5 - 1 or 9. Thus, we have: 52 = 1 + 3 + 5 + 7 + 9 = 25. To form the next square number beyond 25 all we do is add 2n + 1. To get 62we add 2·5 + I, or 11, to 25 resulting in the new square number 36. Of course, by computing the first few sums of odd numbers and seeing that these sums are perfect squares, we do not prove that it is always the case. Could there be some large value of n such that when we add all the odd numbers together up to 2n - 1 we get a number that is not a perfect square?

We use special symbols when adding the numbers of a number sequence into a number series. Sometimes we use a large 5, and if we know the number of terms in the sequence is n, we show the sum as 5 n, meaning that we have added n items. Thus, the sum of the first five numbers in the natural number sequence would be: 55 = 1 + 2 + 3 + 4 + 5 = 15 On other occasions we use the Greek letter sigma or k. Therefore, we can show the above series in several different ways: 55 = L i = 1 + 2 + 3 + 4 + 5 = 15 i=l Below the sigma we have i = 1 which tells us the first term we are adding; and above the sigma we have a 5 which shows us the last or fifth term to be added.

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