Think of at least two examples of each type of numbers

4. Give Russian equivalents to the following words and expressions:

Primality, proper divisor, to establish a result, Dark Ages, to prove a speculation, to demonstrate by factorizing a number, to be divisible by, decimal digits.

5. Answer the questions:

1. Who was the first to study prime numbers?

2. What were the mathematicians of Pythagoras's school mainly interested in?

3. What proof of the Fundamental Theorem of Arithmetic did Euclid give?

4. What period is called the Dark Ages?

5. What does Fermat’s Little Theorem state? Why is it so important?

6. What kind of numbers are called Mersenne numbers and why?

Scientific contribution of what mathematician to the prime numbers theory is described in the following passages? Arrange the passages in the chronological order.

1) He studied numbers of the form 2ⁿ – 1, which nowadays are known as numbers called after him. The largest known prime number is the number of exactly the same form.

2) This mathematician managed to show that all even perfect numbers are of such a form: 2 ⁿ ^-1(2 ⁿ - 1).

3) The proof of the Fundamental Theorem of Arithmetic together with the proof that there are infinitely many prime numbers was given by him

4) Being interested in numbers for their mystical and numerological properties, they understood the idea of primality and were occupied with the study of perfect and amicable numbers.

5) This mathematician devised a new method of factorizing large numbers.

7. Enrich each passage using the information from the text and speak on the following topics:

1) The mathematicians of Pythagoras's school and their scientific work.

2) Euclid’s speculations about prime numbers in his “Book of the Elements”.

3) Fermat’s proof of the Fundamental Theorem of Arithmetic, his Little Theorem and other works.

4) Mersenne numbers.

Retell the text “Prime numbers” using the statements from the previous assignment as the plan.

Text 2

Before you read:

Consider the following unsolved problems in the theory of prime numbers and give accurate translation.

Some unsolved problems

1. The Twin Primes Conjecture that there are infinitely many pairs of primes only 2 apart.

2. Goldbach's Conjecture (made in a letter by C Goldbach to Euler in 1742) that every even integer greater than 2 can be written as the sum of two primes.

3. Are there infinitely many primes of the form n ² + 1?
(Dirichlet proved that every arithmetic progression: { a + bn | n N } with a, b coprime contains infinitely many primes.)

4. Is there always a prime between n ² and (n + 1)²?
(The fact that there is always a prime between n and 2 n was called Bertrand's conjecture and was proved by Chebyshev.)

5. Is there an arithmetic progression of consecutive primes for any given (finite) length? e.g. 251, 257, 263, 269 has length 4. The largest example known has length 10.

6. Are there infinitely many sets of 3 consecutive primes in arithmetic progression?

7. n ² - n + 41 is prime for 0 n 40. Are there infinitely many primes of this form? The same question applies to n ² - 79 n + 1601 which is prime for 0 n 79.