# One-five-three

"Simon Peter went up and drew the net to land full of great fishes, an hundred and fifty and three".

Simon Peter had gone fishing on Lake Tiberias, but at first he caught nothing. Then Jesus appeared on the lakeshore, and the disciple's luck changed. He at once caught 153 fish - a most unusual number. (This was interpreted numerologically, especially by St. Augustine.)

Firstly 153 is a triangle number, the sum of all numbers from 1 to 17, but, more importantly, it is the sum of the cubes of its own digits:
13 + 53 + 33 = 153.

The number 153 may also have appealed to ancient mathematicians because it crops up so often when we calculate with numbers that can be divided by three without remainder. Take 27, for example:
23 + 73 = 351 (153 reversed)

Any other multiple of three will sooner or later give the same result; but you will have to repeat the process of cubing and adding. The number 123 needs 8 of these operations, while 99 needs three:

1. 93 + 93 = 1458
2. 13 + 43 + 53 + 83 = 702;
3. 73 + 03 + 23 = 351.

These are all in base ten - so what about other number bases? Can we find similar patterns?

Here's one in base six : 23 + 43 + 33 = 243.
(243 base six is 99 in base ten - again a multiple of 3. Is it only multiples of 3 that behave this way?)

Try the number 2002 - but change it to base twelve first.

Just to finish, 153 is also the sum of the first five factorial numbers:
153 = 1! + 2! +3! +4! +5!

(If you don't know: a factorial number is the product of the number and all the integers below it down to 1; e.g. 4! = 4 x 3 x 2 x 1 = 24, 5! = 5 x 4 x 3 x 2 x 1 = 120, and so on).

## Results so far

 Base Numbers Equivalent in base ten and base twelve three 122 17 15 four 20, 21,130, 203 8, 9, 28, 35 8, 9, 24, 2E five 103 28 24 six 131, 243, 514, 1055 55, 99, 190, 251 47, 83, 13T, 18E seven 12 9 9 eight 134, 660 92, 432 78, 300 nine 30 27 23 ten 153 153 109 eleven 32, 105, 307, 708, T06 35, 126, 370, 855, 1216 2E, T6, 26T, 5E3, 854 twelve T83, 11TT 1539, 2002 T83, 11TT

(Bases higher than twelve not tried yet)

T stands for ten and E for eleven in bases higher than ten.