The Bible tells of the Apostles going fishing and catching exactly 153 fish. It so happens that 153 is a "triangular" number (in the Pythagorean sense), being the sum of the first 17 integers. It's also the sum of the first five factorials. A slightly more obscure property of 153 is that it equals the sum of the cubes of its decimal digits. In fact, if we take ANY integer multiple of 3, and add up the cubes of its decimal digits, then take the result and sum the cubes of its digits, and so on, we invariably end up with 153. For example, since the number 4713 is a multiple of 3, we can reach 153 by iteratively summing the cubes of the digits, as follows: starting number = 4713 4^3 + 7^3 + 1^3 + 3^3 = 435 4^3 + 3^3 + 5^3 = 216 2^3 + 1^3 + 6^3 = 225 2^3 + 2^3 + 5^3 = 141 1^3 + 4^3 + 1^3 = 66 6^3 + 6^3 = 432 4^3 + 3^3 + 2^3 = 99 9^3 + 9^3 = 1458 1^3 + 4^3 + 5^3 + 8^3 = 702 7^3 + 2^3 = 351 3^3 + 5^3 + 1^3 = 153 <----- The fact that this works for any multiple of 3 is easy to prove. First, recall that any integer n is congruent modulo 3 to the sum of its decimal digits (because the base 10 is congruent to 1 modulo 3). Then, letting f(n) denote the sum of the cubes of the decimal digits of n, by Fermat's little theorem it follows that f(n) is congruent to n modulo 3. Also, we can easily see that f(n) is less than n for all n greater than 1999. Hence, beginning with any multiple of 3, and iterating the function f(n), we must arrive at a multiple of 3 that is less than 1999. We can then show by inspection that every one of these reduces to 153. Since numerology has been popular for thousands of years, it's conveivable that some of the special properties of the number 153 might have been known to the author of the Gospel. Of course, our modern decimal number system wasn't officially invented until much later, so it might seem implausible that the number 153 was selected on the basis of any properties of its decimal digits. On the other hand, the text (at least in the English translations) does specifically state the number verbally in explicit decimal form, i.e., "Simon Peter went up, and drew the net to land full of great fishes, an hundred and fifty and three: and for all there was so many, yet was not the net broken." John, 21:11 Thus, rather than talking about scores or dozens, it speaks in multiples of 100, 10, and 1. Since only multiples of 3 reduce to 153, we might ask what happens to the other numbers. It can be shown that all the integers congruent to 2 (mod 3) reduce to either 371 or 407. The integers congruent to 1 (mod 3) reduce to one of the fixed values 1 or 370, or else to one of the cycles [55, 250, 133], [160, 217, 352], [136, 244], [919, 1459]. Within the congruence classes modulo 3 there doesn't seem to be any simple way of characterizing the numbers that reduce to each of the possible fixed values or limit cycles. Naturally we could perform similar iterations on the digits of a number in any base. One of the more interesting cases is the base 14, in which 2/3 of all number eventually fall into a particular cycle. Coincidentally, this cycle includes the decimal number 153, but it also includes 26 other numbers, for a total length of 27, which is 3 cubed (which the mystically minded should have no trouble associating with the Trinity). The decimal values of this base-14 cycle are 9 729 1028 368 1793 738 2027 2395 1756 2925 3926 433 2213 1396 1344 1944 4185 2605 2262 2186 1347 1971 2331 3402 153 3197 198 Again, there doesn't appear to be any way of distinguishing the numbers that reduce to this cycle from those that don't, other than by performing the iterations. By considering sums of higher powers (or polynomials) of the digits in other bases, we can produce a wide variety of arbitrarily long (but always finite) cycles. The number 153 is also sometimes said to be related to a symbol called the "vesica piscis", which consists of the intersection of two equal circles whose centers are located on each others circumferences. However, the relevance of the number 153 to this shape is rather dubious. It rests on the fact that the ratio of the length to the width of this shape equals the square root of 3, and one of the convergents of the continued fraction for the square root of 3 happens to be 265/153. It is sometimes claimed that this was the value used by Archimedes, but this is only partly true. Archimedes knew that the square root of 3 is irrational, and he determined that its value lies between 265/153 and 1351/780, the latter being another convergent of the continued fraction.

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