Write a program to perform the four basic arithmetic operations, +, –, ·, and /, on very l...

Write a program to perform the four basic arithmetic operations, +, –, ·, and /, on very large integers; the result of division should also be an integer. Apply these operations to compute 12345, or the hundredth number in the sequence 1 * 2 + 3, 2 * 32 + 4, 3 * 43 + 5, . . . . Also apply them to compute the Gödel numbers of arithmetical expressions.

The Gödel numbering function GN first establishes a correspondence between basic elements of language and numbers:

where S is the successor function. Then, for any formula F = s1s2 … sn:

GN(´s1s2 . . . sn´) = 2GN(s1) * 3GN(s2) * … * pnGN(sn)

where pn is the nth prime. For example,

GN(1) = GN(S0) = 210 * 39

and

GN(´x1 + x3 = x4´) = 211+2 * 32 * 511+6 * 71 * 1111+8

In this way, every arithmetic expression can be assigned a unique number. This method has been used by Gödel to prove theorems, known as Gödel’s theorems, which are of extreme importance for the foundations of mathematics.