# Open AI III, See part I and See part II

A bit more experiments Open AI.

```Who is the most famous important mathematician

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.

included a list of mathematicians.
```
Lets look up some local knowledge:
```
Who us currently the chair of the Harvard mathematics department

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

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?

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.

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

The solution of the equation is x = ± 0.73908513321516064.

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

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)

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

```
```Find the anti derivative of sin(x)^3

cos(x)^3/3

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

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.

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?

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.

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],f]
```
```How many quadratic residues are there modulo 7

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

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

There are 50 quadratic residues modulo 101

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.

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.

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)

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.