Convergence to the Number 153 from any Positive Number

There are three ruling ideas, three so to say, spheres of thought, which pervade the whole body of mathematical science, to some one or other of which, or to two or all three of them combined, every mathematical truth admits of being referred; these are the three cardinal notions, of Number, Space and Order. Arithmetic has for its object the properties of number in the abstract. In algebra, viewed as a science of operations, order is the predominating idea. The business of geometry is with the evolution of the properties of space, or of bodies viewed as existing in space.

J. J. Sylvester

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Mathematics is all about numbers, and one important number is 153. 153 is the natural number following 152 and preceding 154. In this blog post, let’s play with the number 153. Yes, we’ll play a game with the number 153 too.

The divisors of 153 are 1, 3, 9, 17, 51 and 153. The sum of divisors of 153 is 1 + 3 + 9 + 17 + 51 + 153 = 234 and the sum of aliquot divisors (any divisor excluding the number itself) of 153 is 1 + 3 + 9 + 17 + 51 = 81 = 92, a perfect square.

153 is a triangular number and it is the sum of the first 17 integers. That is, 153 = 1 + 2 + 3 + ⋅⋅⋅ + 17. Also, 153 is the sum of the first five positive factorials, i.e., 153 = 1! + 2! + 3! + 4! + 5!.

Since 153 = 13 + 53 + 33, it is a 3-Narcissistic number. It is the smallest three-digit number which can be expressed as the sum of cubes of its digits.

Sum of the digits of 153 is = 1 + 5 + 3 = 9 and it is divisible by the sum of its digits: 153/(1 + 5 + 3) = 17.

A Game with 153

The number 153 is indeed interesting. From any positive integer (one, two, three, four or more digits), divisible by 3, following certain rules, we can always get 153. What are the rules? They are given below.

Rules for obtaining the number 153

  • (a) Choose a number that is divisible by 3. Let the number be xyz…;
  • (b) Extract the digits (in base 10) of the number. The digits are x, y, z, . . .;
  • (c) Take the sum of their cubes. That is, find x3 + y3 + z3 + ⋅⋅⋅;
  • (d) If this yields 153, stop;
  • (e) Otherwise, repeat the procedure.

Let’s see the rule in action.

Example 1. Consider the singledigit number 3.

Clearly, the only digit of this number is 3 and 33 = 27.

Next, the digits of the last number are 2, 7 and 23 + 73 = 351.

Finally, the digits of the last number are 3, 5, 1 and 33 + 53 + 13 = 153.

Here, three steps are required to reach 153.

Note. In this process, if we get any three-digit number comprising the digits 1, 3 and 5 only, we are done. For example, if we proceed one step further, then we have 13 + 53 + 33 = 153.

Example 2. Consider the twodigit number 84.

The digits of this number are 8 and 4. Now,

83 + 43

= 512 + 64

= 576.

The digits of the last number are 5, 7 and 6. We have,

53 + 73 + 63

= 125 + 343 + 216

= 684.

Again, the digits of the last number are 6, 8 and 4. We have,

63 + 83 + 43

= 216 + 512 + 64

= 792.

Now, the digits of the last number are 7, 9 and 2. We have,

73 + 93 + 23

= 343 + 729 + 8

= 1080.

The digits of the last number are 1, 0, 8 and 0. We have,

13 + 03 + 83 + 03

= 1 + 0 + 512 + 0

= 513.

Finally, the digits of the last number are 5, 1 and 3. We have,

53 + 13 + 33

= 125 + 1 + 27

= 153.

Only six steps are required here to reach 153.

Example 3. Consider the threedigit number 177.

Following the similar lines of argument as above, we obtain

Step 1 : 13 + 73 + 73 = 687

Step 2 : 63 + 83 + 73 = 1071

Step 3 : 13 + 03 + 73 + 13 = 345

Step 4 : 33 + 43 + 53 = 216

Step 5 : 23 + 13 + 63 = 225

Step 6 : 23 + 23 + 53 = 141

Step 7 : 13 + 43 + 13 = 66

Step 8 : 63 + 63 = 432

Step 9 : 43 + 33 + 23 = 99

Step 10 : 93 + 93 = 1458

Step 11 : 13 + 53 + 53 + 83 = 702

Step 12 : 73 + 03 + 23 = 351

Step 13 : 33 + 53 + 13 = 153

Here, thirteen steps are required to reach 153.

How this works? Can you think about a formal proof?

Your suggestions are eagerly and respectfully welcome! See you soon with a new mathematics blog that you and I call Math1089 – Mathematics for All!“.

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