Catch of the Day (153 Fishes)

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.