Open AI III, See part I and See part II

A bit more experiments Open AI.

1. About Mathematicians

Who is the most famous important mathematician

AI Answer:
One of the most famous and important mathematicians of 
all time is Isaac Newton. He is best known for his work in 
calculus, optics, gravity and motion, and is the founder 
of modern physics. He is widely considered to be one of 
the most influential scientists in history.

Grading:
This is just your opinion, man. A better answer would have
included a list of mathematicians. 
Lets look up some local knowledge:
 
Who us currently the chair of the Harvard mathematics department

AI Answer:
The current chair of the Harvard Mathematics Department is 
Professor Yau Shing-Tung.

Grading: 
Wrong. That was more than 10 years ago. Funny that it would
put Yau before Shing-Tung. 
And now some opinion:
Who is the most famous Harvard Mathematician?

AI Answer:
The most famous Harvard mathematician is probably Benoit Mandelbrot, 
who was a French-American mathematician and polymath with broad interests 
in the practical sciences, especially regarding what he labeled as 
"the art of roughness" of physical phenomena and "the uncontrolled element 
in life". He is best known as the father of fractal geometry.

Grading: 
Mandelbrot had indeed been visiting Harvard but he was never 
Harvard professor. He is undoubtably famous. 

2. Numerics

Find the solution of the equation cos(x)=x

AI Answer:
The solution of the equation is x =  0.73908513321516064.

Grading: 
The positive solution is ok. The negative solution is false. 
The graph of cos(x) and x only intersect in 0.739 ....

3. Some calculus

Find the anti derivative of sin(x)^10

AI Answer:
pi/11 sin(x)^11 + 5/11 cos(x) sin(x)^10 + (1/2) sin(2x) sin(x)^9 + 2/33 cos(3x) sin(x)^8 + 
1/44 sin(4x) sin(x)^7 + 5/165 cos(5x) sin(x)^6 + (1/7) sin(6x) sin(x)^5 + 2/77 cos(7x) sin(x)^4 + 
1/88 sin(8x) sin(x)^3 + 5/323 cos(9x) sin(x)^2 + (1/10) sin(10x) sin(x) + 2/99 cos(11x) sin(x) + 1/11 sin(12x)

Grading: 
Total nonsense. The answer is 

         63 x   105 Sin[2 x]   15 Sin[4 x]   15 Sin[6 x]   5 Sin[8 x]   Sin[10 x]
       = ---- - ------------ + ----------- - ----------- + ---------- - ---------
         256        512            256          1024          2048        5120

What about something simpler?
Find the anti derivative of sin(x)^3

AI Answer: 
cos(x)^3/3 

Grading: 
Total nonsense. 
Some opinion
What is the most important theorem in multi-variable calculus?

AI Answer:
The Fundamental Theorem of Calculus is widely considered to be the 
most important theorem in multi-variable calculus. This theorem states that the 
integral of a function over a region is equal to the sum of the integrals of 
the function over each subregion of the region. This theorem is essential for 
understanding the relationship between derivatives and integrals, and 
is used in a variety of contexts in mathematics, physics, and engineering.


Grading: 
Very poor answer. The program stated an addition rule for integrals and not
the fundamental theorem of calculus

4. Some number theory

Which primes can be written as a sum of two squares?

AI Answer:
The primes that can be written as a sum of two 
squares are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 
31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 
83, 89, 97.

Grading: 
Total nonsense. The program just listed all primes
smaller than 100. The answer is all primes of the 
form 4k+1: here is a program computing the first 30

f[x_]:=Mod[x,4]==1;Select[Prime[Range[30]],f]
How many quadratic residues are there modulo 7

AI Answer:
There are four quadratic residues modulo 7: 
1, 2, 4 and 2.

Grading:
The answer 4 is correct if one includes 0, but the number 2 has been included
twice. The quadratic residues are 0,2,3,4. 
But the program is confused on whether to include 0 as a quadratic residue:
How many quadratic residues are there modulo 101

AI Answer:
There are 50 quadratic residues modulo 101

Grading:
This is correct if one does not include 0 as a quadratic residue. But usually
one does include 0. 

5. Some geometry

Prove Pythagoras theorem.

AI Answer:
Pythagoras theorem states that in a right triangle, the square of the hypotenuse 
(the side opposite the right angle) is equal to the sum of the squares of the 
other two sides.

Proof:
Let ABC be a right triangle, with the right angle at C.
Draw squares on each of the sides AB, BC, and AC.
Since ABC is a right triangle, the square on the hypotenuse AC is 
the sum of the squares on the sides AB and BC.  Therefore, AC^2 = AB^2 + BC^2 QED.

QED.

Grading:
The statement of the theorem is correct. The proof just uses the theorem
and is therefore not valid. 
And here a basic multi-variable geometry problem:
What is the distance between the line r(t) = (t,t,t) and the point P = (3,4,5)

AI Answer:

The distance between the line and the point is sqrt(6). 
This can be calculated using the formula: 
distance = sqrt{(3-t)^2 + (4-t)^2 + (5-t)^2}
Setting t = 3, we get: sqrt{6}

Grading: This is incorrect. The distance can be computed as the area/base
of a parallelogram which is 
|[3,4,5] x [1,1,1]|/|[1,1,1]| =[[-1,2,-1]|/|[1,1,1]|=sqrt(2)
Formulate Morley's theorem.

AI Answer:
Morley's theorem states that in any triangle, the three angle bisectors, 
the three medians, and the three altitudes all intersect at a single point.

Grading:
These are all correct results but this is not Morley's theorem. Morley's theorem
deals with angle trisectors.