By Jonathan M. Borwein

Thirty years in the past mathematical, in place of utilized numerical, computation was once tough to accomplish and so particularly little used. 3 threads replaced that: the emergence of the private computing device; the invention of fiber-optics and the resultant improvement of the fashionable net; and the construction of the 3 “M’s” Maple, Mathematica and Matlab.

We intend to cajole that Mathematica and different related instruments are worthy realizing, assuming simply that one needs to be a mathematician, a arithmetic educator, a working laptop or computer scientist, an engineer or scientist, or someone else who wishes/needs to exploit arithmetic larger. We additionally desire to provide an explanation for the way to develop into an "experimental mathematician" whereas studying to be greater at proving issues. to complete this our fabric is split into 3 major chapters by way of a postscript. those hide ordinary quantity concept, calculus of 1 and a number of other variables, introductory linear algebra, and visualization and interactive geometric computation.

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**Example text**

Now suppose that a ≤ n. It must, necessarily, be the case that b ≥ n, because √ √ √ b ≤ k =⇒ ab < n n = n √ √ which would contradict a and b being codivisors. Similarly, if a ≥ n then√b ≤ n. It follows, then, that once we’ve found all the divisors less than or equal to n then we have also found all the larger divisors in the codivisors. We can use this fact to modify our loop and make it a little more eﬃcient. There is a small problem here, however, because the square root of a number is not always a natural number.

To illustrate this idea, let’s look at some related notions to divisors and perfect numbers. Observe that if a number n is not perfect, then the sum of the proper divisors is either strictly greater, or strictly less than n itself. If the sum of the proper divisors is less than n we say the number is deﬁcient, and if the sum of the proper divisors is greater than n then we say that n is abundant. We now have three mutually exclusive options for any natural number. This idea lends itself nicely to a nested decision.

However, we may need to be a little speciﬁc in what we ask Mathematica. In[103]:= Abs[a[N + 1] / a[N]] < 1 Out[103]= Abs N2 <1 2+N The less than operator (<) should have produced an output of either True or False. The fact that it simply re-iterated the input tells us that Mathematica does not know the answer. With a bit of thought this is hopefully not too surprising. 2 Putting It Together 17 the question doesn’t make any sense for N = −1, and the statement is false for N < −1. As observed above, however, we’re only interested in strictly positive values of N .