Famous Math Problems/1

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This video is by Dr. Norman Wildberger, professor (emeritus) at the University of New South Wales. He introduces the series in this video, and establishes a scale of difficulty ranking. Finally, he discusses his aims in the series.

Factoring Large Numbers
The first problem is an example of an impossible problem:

Factor $$z=10^{{{{{10}^{10}}^{10}}^{10}}^{10}}+23$$ into the product of primes.

Recall that prime numbers are those which can only be written as the product of themselves and 1. Euclid proved that there is no last prime number.

Fundamental Theory of Arithmetic
Euclid said ever Natural Number can be written uniquely as the product of powers of primes. But this begs the question, what is a valid number?

Possible Steps
When faced with an impossible problem, it is wise to break it into pieces. Various powers of 10 + 23 all have 3 as a factor. But is there an overall pattern to these?

Tools
The tools of problems like this are organized on the rubric "Modular Arithmetic". Within that field of study, the most important result is Fermat's Little Theorem (which was generalized by Euler). Prof W then shows that z is not divisible by 7.

There are simpler versions of this problem which are solvable. Unforunately, most numbers less than z canot be written in our universe.