Aristotle also claimed that women have fewer teeth than men. He was never curious to see if this easy -to-disprove claim was true. If only he asked his wife directly to open her mouth so he can count her teeth, his entire view on women might have changed.

It seems that much of today's Higher Algebra research is Aristotle-esque in that it focuses on categorizing abstract structures, often without visualizing or explicitly calculating their contents.

For example, I found out that some people have a hard time visualizing what would happen if they put every element

**a**smaller than

**b**to the power of every element

**x**smaller than

**b**in a consecutive order, and then make the results restricted to modulo

**b,**and then arrange these results in a consecutive manner in a table like so:

Z sub 35 under exponentiation |

This table helps visualize the exact position of phi(n) within the first non - trivial row. Since in this case n is composed of 2 distinct odd primes, the exact position of phi(n)/2 is also revealed. It does not matter what type of an abstract structure it represents, all that matters is the fact that the position of phi(n)/2 is easily revealed.

I suspect that people who are confused by this exponentiation table are confused because they expect to see at least one row k where k is coprime with 35 that lists all integers of Z sub 35 just like in the multiplication table below but there aren't any in the exponentiation table when n is composed of 2 distinct odd primes because it is not the right abstract algebraic structure for this, namely it is not cyclic under exponentiation.

Z sub 35 under multiplication |

I picked 5 and 7 from the list of "safe primes" below but it works when n is the product of any two primes both greater than 2.

5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907,...

I also picked 5 and 7 because their product is 35, which is the absolute maximum number of integers that a regular piece of graph paper can accommodate but it is absolutely irrelevant which two distinct odd primes are chosen. What matters is what happens when each integer smaller than the product of two distinct odd primes is put to the power of another integer smaller than the product of primes in a consecutive order.