Examples of TEX

• For any real number θ, we have

sin^{2}(θ) + cos^{2}(θ) = 1.

• Riemann zeta function is defined by

ζ(s) = ∑_{n=1}^{∞} \frac{1}{n^{s}} = 1 + \frac{1}{2^{s}} + \frac{1}{3^{s}} + \frac{1}{4^{s}} + · · · .

Fermat’s Last Theorem. There are no positive integer solutions x, y, z of the equation

x^{n} + y^{n} = z^{n}  (1.1)

for integers n ≥ 3.

• Using software Maple, we can calculate factorial of 100:

100! = 6933262154439441526816992388562667004907159682643816214\

6859296389521759999322991560894146397615651828625369792\

0827223758251185210916864000000000000000000000000

= 2^{97} ∗ 3^{48} ∗ 5^{24} ∗ 7^{16} ∗ 11^{9} ∗ 13^{7} ∗ 17^{5} ∗ 19^{5} ∗ 23^{4} ∗ 29^{3} ∗ 31^{3} ∗ 37^{2}

× 41^{2} ∗ 43^{2} ∗ 47^{2} ∗ 53 ∗ 59 ∗ 61 ∗ 67 ∗ 71 ∗ 73 ∗ 79 ∗ 83 ∗ 89 ∗ 97

• Numbers of the form F_{n} = 2^{2^{n}} +1 are called Fermat numbers. Fermat conjectured that F_{n} is prime for every n ≥ 0. For n = 0, …, 4 the numbers F_{n} = 3, 5, 17, 257, 65537 are indeed prime, but in 1732 Euler showed that the next Fermat number

F_{5} = 2^{2^{5}} + 1 = 4294967297 = 641 × 6700417

is composite.

• The number 1 + \sqrt{5} can be written as

1 + \sqrt{5} = 3 + \frac{1}{4 + \frac{1}{4 + \frac{1}{4 + \frac{1}{4 + ⋯}}}} .