Lesson test: Why prove anything

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Why prove anything

1.[2p]
Euler's expression $n^{2} + n + 41$ is prime for every $n$ from $0$ to $39$ . Why does it fail at $n = 40$ ?

$40^{2} + 40 = 40 \times 41$ , so the value is $40 \times 41 + 41 = 41^{2}$

Correct

$40$ is even, so the value is even

The answer is: $40^{2} + 40 = 40 \times 41$ , so the value is $40 \times 41 + 41 = 41^{2}$

The value $1681$ is divisible by $3$

The answer is: $40^{2} + 40 = 40 \times 41$ , so the value is $40 \times 41 + 41 = 41^{2}$

The value is prime, and the first failure is at $n = 41$

The answer is: $40^{2} + 40 = 40 \times 41$ , so the value is $40 \times 41 + 41 = 41^{2}$

2.[1p]
Six points on a circle, joined by every chord with no three chords meeting at one interior point, cut the disc into $32$ regions.

True

The answer is: False

False

Correct

3.[1p]
What is the smallest $n$ greater than $1$ for which Pólya's conjecture fails, found by Minoru Tanaka in 1980?

906,150,257

Correct

1,681

The answer is: 906,150,257

1,601

The answer is: 906,150,257

No counterexample has been found, only proved to exist

The answer is: 906,150,257

4.[2p]
Which of these may justify a line of a proof?

Select all that apply

A definition

Correct

An axiom

Correct

A result that has already been proved

Correct

A check of the first million cases

The answer is: A definition, An axiom, A result that has already been proved

A carefully drawn diagram

The answer is: A definition, An axiom, A result that has already been proved

5.[3p]
Match each term to its definition.

Even integer
Odd integer
Prime
Composite

greater than $1$ , with only $1$ and itself as positive divisors
a product of two integers each greater than $1$
equal to $2 k + 1$ for some integer $k$
equal to $2 k$ for some integer $k$

Show the answer

Even integer: equal to $2 k$ for some integer $k$ Odd integer: equal to $2 k + 1$ for some integer $k$ Prime: greater than $1$ , with only $1$ and itself as positive divisors Composite: a product of two integers each greater than $1$

6.[1p]
By the definition of even, $0$ is even.

True

Correct

False

The answer is: True

7.[2p]
Put the lines of this proof that the sum of two even integers is even in order.

Let $a$ and $b$ be even integers.
Since $k$ and $m$ are integers, so is $k + m$ .
Then $a + b = 2 k + 2 m = 2 (k + m)$ .
So $a + b$ is twice an integer, and by the definition of even, it is even.
By the definition of even, $a = 2 k$ and $b = 2 m$ for some integers $k$ and $m$ .

Show the answer

a, b, c, d, e

8.[2p]
A proof that the sum of two odd integers is even begins "Let $a = 2 k + 1$ and $b = 2 k + 1$ ." What is wrong with it?

Using the same $k$ for both forces $a = b$ , so it proves the claim only for a number added to itself

Correct

Odd integers must be written $2 k - 1$ , not $2 k + 1$

The answer is: Using the same $k$ for both forces $a = b$ , so it proves the claim only for a number added to itself

Nothing, since $k$ is arbitrary

The answer is: Using the same $k$ for both forces $a = b$ , so it proves the claim only for a number added to itself

A proof may not use letters for integers it has not yet found

The answer is: Using the same $k$ for both forces $a = b$ , so it proves the claim only for a number added to itself

9.[3p]
Which of these statements about evidence and proof are correct?

Select all that apply

A single counterexample refutes a claim about every natural number

Correct

Checking a claim for nine hundred million cases proves it for all cases

The answer is: A single counterexample refutes a claim about every natural number, One proof can establish infinitely many cases at once, A proof's conclusions hold only under the assumptions it starts from

One proof can establish infinitely many cases at once

Correct

A proof's conclusions hold only under the assumptions it starts from

Correct

10.[1p]
Seven points on a circle are joined by every chord, with no three chords meeting at one interior point. How many regions do the chords cut the disc into?

CorrectNot quite: 57

11.[2p]
What is unusual about the refutation of the Mertens conjecture, that $| M (n) | < \sqrt{n}$ for every $n > 1$ ?

It was proved false in 1985, yet no single $n$ that breaks it is known

Correct

It was refuted by a computer search that found the first counterexample

The answer is: It was proved false in 1985, yet no single $n$ that breaks it is known

It was found to be false only for negative $n$

The answer is: It was proved false in 1985, yet no single $n$ that breaks it is known

It turned out to be true after all, once the checks went far enough

The answer is: It was proved false in 1985, yet no single $n$ that breaks it is known

12.[1p]
Checking by multiplication that $59 \times 509 = 30 031$ is a complete proof that $30 031$ is not prime.

True

Correct

False

The answer is: True

13.[3p]
Match each word to the role of the statement it names.

Axiom
Lemma
Corollary
Conjecture

follows with little extra work from a theorem just proved
accepted without proof, as a starting point
proved mainly to help prove something else
believed by someone, but not yet proved

Show the answer

Axiom: accepted without proof, as a starting point Lemma: proved mainly to help prove something else Corollary: follows with little extra work from a theorem just proved Conjecture: believed by someone, but not yet proved

14.[2p]
The proof from the axioms that $a \times 0 = 0$ begins by writing $a \times 0 = a \times (0 + 0)$ . Which axiom turns that into $a \times 0 + a \times 0$ ?

The distributive law, $a (b + c) = a b + a c$

Correct

Commutativity, $a b = b a$

The answer is: The distributive law, $a (b + c) = a b + a c$

The existence of negatives, $a + (- a) = 0$

The answer is: The distributive law, $a (b + c) = a b + a c$

The identity $a \times 1 = a$

The answer is: The distributive law, $a (b + c) = a b + a c$

1.[2p] Euler's expression n2+n+41 is prime for every n from 0 to 39. Why does it fail at n=40?

2.[1p] Six points on a circle, joined by every chord with no three chords meeting at one interior point, cut the disc into 32 regions.

3.[1p] What is the smallest n greater than 1 for which Pólya's conjecture fails, found by Minoru Tanaka in 1980?

4.[2p] Which of these may justify a line of a proof?

5.[3p] Match each term to its definition.

6.[1p] By the definition of even, 0 is even.

7.[2p] Put the lines of this proof that the sum of two even integers is even in order.

8.[2p] A proof that the sum of two odd integers is even begins "Let a=2k+1 and b=2k+1." What is wrong with it?

9.[3p] Which of these statements about evidence and proof are correct?

10.[1p] Seven points on a circle are joined by every chord, with no three chords meeting at one interior point. How many regions do the chords cut the disc into?

11.[2p] What is unusual about the refutation of the Mertens conjecture, that |M(n)|<n for every n>1?

12.[1p] Checking by multiplication that 59×509=30031 is a complete proof that 30031 is not prime.

13.[3p] Match each word to the role of the statement it names.

14.[2p] The proof from the axioms that a×0=0 begins by writing a×0=a×(0+0). Which axiom turns that into a×0+a×0?

1.[2p]
Euler's expression $n^{2} + n + 41$ is prime for every $n$ from $0$ to $39$ . Why does it fail at $n = 40$ ?

2.[1p]
Six points on a circle, joined by every chord with no three chords meeting at one interior point, cut the disc into $32$ regions.

3.[1p]
What is the smallest $n$ greater than $1$ for which Pólya's conjecture fails, found by Minoru Tanaka in 1980?

4.[2p]
Which of these may justify a line of a proof?

5.[3p]
Match each term to its definition.

6.[1p]
By the definition of even, $0$ is even.

7.[2p]
Put the lines of this proof that the sum of two even integers is even in order.

8.[2p]
A proof that the sum of two odd integers is even begins "Let $a = 2 k + 1$ and $b = 2 k + 1$ ." What is wrong with it?

9.[3p]
Which of these statements about evidence and proof are correct?

10.[1p]
Seven points on a circle are joined by every chord, with no three chords meeting at one interior point. How many regions do the chords cut the disc into?

11.[2p]
What is unusual about the refutation of the Mertens conjecture, that $| M (n) | < \sqrt{n}$ for every $n > 1$ ?

12.[1p]
Checking by multiplication that $59 \times 509 = 30 031$ is a complete proof that $30 031$ is not prime.

13.[3p]
Match each word to the role of the statement it names.

14.[2p]
The proof from the axioms that $a \times 0 = 0$ begins by writing $a \times 0 = a \times (0 + 0)$ . Which axiom turns that into $a \times 0 + a \times 0$ ?