Angličtina pro matematiky III
Course materials and homework week IX.
Terence Tao: Structure and Randomness in the Prime Numbers, UCLA 3:00 – 8:20
Decide whether the following statements are true or false.
1) Dr. Tao´ s subject of research is prime No.
2) This topic is 200 years old.
3) There is still much work to be done concerning prime numbers.
4) Natural numbers are similar to elements in physics.
5) One of the first theorems about primes goes back to the ancient Greece.
6) There is only one way of expressing 50 as a multiplication of primes.
7) The product of primes cannot be rearranged.
8) 1 cannot be split into smaller No, therefore the Fundamental theorem of arithmetic is not true.
9) Euclid believed that the number of primes is infinite.
10) There are less prime numbers than atomic elements.
11) Reduction ad absurdum means a proof by confirmation.
12) You prove something, then pretend it is false, and then show that it can happen which indirectly shows it is false.
Fill in the missing words in the proof:
Euclid´ s proof is the classic example of reduction ad absurdum.
- Suppose, for sake of contradiction, that there were only finitely many prime numbers p1,
p2…..pn (e.g. suppose 2,3,5 were ????????? primes).
- Multiply all the primes together and add (or ??????) 1: P=p1 p2………pn + 1. (e.g.
P=2x3x5 + 1=?? or 31.)
- Than P is a ???????? number larger than 1 but P is not ??????? by any of the prime numbers.
- This contradicts the fundamental theorem of arithmetic. Hence there are infinitely many primes.
Terence Tao: Structure and Randomness in the Prime Numbers, UCLA 3:00 – 8:20
Decide whether the following statements are true or false.
13) Dr. Tao´ s subject of research is prime No. F number theory
14) This topic is 2 000 years old. F 2000
15) There is still much work to be done concerning prime numbers. T
16) Natural numbers are similar to elements in physics. F chemistry
17) One of the first theorems about primes goes back to ancient Greece. T
18) There is only one way of expressing 50 as a multiplication of primes. T
19) The product of primes cannot be rearranged. F
20) 1 cannot be split into smaller No, therefore the Fundamental Theorem of Arithmetic is not true. F is true
21) Euclid believed that the number of primes is infinite. T
22) There are less prime numbers than atomic elements. F a.e. 26
23) Addition and multiplication are the same as far as primes are concerned. F
24) Reduction ad absurdum means a proof by confirmation. F contradiction
25) You prove something, than pretend it is false, and then show that it can happen which indirectly shows it is false. F true
26) The British mathematician used an example of a chess player or a gambler who offers a king to win the game. F a queen
Fill in the missing words in the proof:
Euclid´ s proof is the classic example of reduction ad absurdum.
- Suppose, for sake of contradiction, that there were only finitely many prime numbers p1,
p2…..pn (e.g. suppose 2,3,5 were the only primes).
- Multiply all the primes together and add (or subtract) 1: P=p1 p2………pn + 1. (e.g.
P=2x3x5 + 1=29 or 31.)
- Than P is a natural number larger than 1 but P is not divisible by any of the prime numbers.
- This contradicts the fundamental theorem of arithmetic. Hence there are infinitely many primes.