What 5tuple corresponds to the number 256 under this type of

What 5-tuple corresponds to the number 256 under this type of encoding?

The Notation <x, y> . We define <x, y > = 2^x( 2y+ 1) -1 . Recall from an earlier example that this function is one-to-one from the set of ordered pairs of natural numbers onto the set of natural numbers. The notation <x,y > is suggestive of an ordered pair, but remember, it\'s a natural number.You can think of this number as \"encoding\" the ordered pair (x, y).

It is easy to see that the function <x, <y, z > >,  which we write more simply as <x, y, z > is a one-to-one function from the set of ordered triples of natural numbers onto the set of natural numbers. It can be used to encoded ordered-triples. Similar devices could be used to encode 4-tuples, 5-tuples, etc.

Example 4. What ordered triple corresponds to the number 43.

We are to find  a, b, and c such that   43 = <a, <b, c > > .

Using  d to denote the number <b, c > ,  we have 43 = <a, d>.

We first calculate  a and d ,   then later we extract  b, c from d.

43 = 44 - 1 = 2⁵ + 2³ + 2² - 1 =   2²(2³+ 2¹ + 2⁰) -1   =   2²(2 ×5+ 1) - 1 = 2^a(2×d +1) -1 .

From this we see that  a = 2 and d = 5

Now we extract  b, c from d.

d = 5 = 6 - 1 = 2² + 2¹  - 1 = 2¹(2¹ + 2⁰) -1 = 2¹( 2 ×1 + 1) -1 =   2^b( 2 ×c + 1) -1

From this we see b = 1 and c = 1 so

43 = <2, <1, 1 > > .

So the answer is the ordered triple   (2, (1, 1)).

Solution

Using  d  to denote the number <b, c > ,  we have 43 = <a, d>.

We first calculate  a and d ,   then later we extract  b, c  from d.

43 = 44 - 1 = 25 + 23 + 22 - 1 =   22(23+ 21 + 20) -1   =   22(2 ×5+ 1) - 1 = 2a(2×d +1) -1 .

From this we see that  a = 2 and d = 5

Now we extract  b, c from d.

d = 5 = 6 - 1 = 22 + 21  - 1 = 21(21 + 20) -1 = 21( 2 ×1 + 1) -1 =   2b( 2 ×c + 1) -1

From this we see b = 1 and c = 1 so

43 = <2, <1, 1 > >