Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- From Year 7: is 91 a prime number?
Answer
No. 91 = 7 × 13. - From Year 8: work out n² + n when n = 6.
Answer
42, because 36 + 6 = 42. - From Year 9: a statement says “every square number is even”. Give one value that shows it is false.
Answer
9, because 9 is a square number and it is odd. Any odd square works.
Words used in this lesson
- conjecture
- a statement thought to be true but not yet proved. For example: every n works.
- counterexample
- one case where a general statement is false. For example: 21 breaks “every odd number above 1 is prime”.
- evidence
- cases that support a claim without proving it. For example: ten cases hold.
- proof
- an argument that covers every case. For example: uses a letter.
- prime
- a whole number greater than 1 with exactly two factors. For example: 17.
- generalization
- a statement about every case, formed from specific cases. For example: the rule.
A rule that keeps working
A maths club in Accra is given a rule: take a whole number n, work out n² + n + 17. For n = 1 the rule gives 19, for n = 2 it gives 23, and both are prime. Someone says: “It always gives a prime.”
Predict first: for how many values of n in a row do you think the rule keeps giving primes? Write a number and a reason, then slide n along the strip in the figure and check each value.
Expression 1, n² + n + 17, at n = 1 gives 19, which is prime. Tested so far: n = 1 only, and every value is prime.
Show me the steps
- Expression 1, n = 1: 1 + 1 + 17 = 19, which is prime. Only one case has been tested.
- Expression 1, n = 6: 36 + 6 + 17 = 59, which is prime. Six cases tested, all prime.
- Expression 2, n = 1: 2 + 11 = 13, which is prime. Slide n up one value at a time and watch the strip.
- Move n along either expression until a square changes colour. That value is not prime, so it is a counterexample, and the statement “always prime” is false.
A conjecture is a statement about every case that has not been proved. Testing cases makes it more convincing, but no number of cases proves it. One counterexample, a single case where the statement fails, proves it false.
So a good test looks for a case that might fail: small values, large values and special values such as 0, 1, negative numbers and primes. And to claim something is true for all n you need a proof, not more examples.
Two common mistakes. Treating a long run of successes as a proof: the rule above works for many cases and still fails. And “disproving” with a case that does not meet the conditions: if the claim is about prime n, a composite n is not a counterexample.
Worked examples
1. Test: “n² + 3 is divisible by 4 for every odd n.”
- n = 1 gives 1 + 3 = 4, divisible by 4. n = 3 gives 9 + 3 = 12, divisible by 4. n = 5 gives 25 + 3 = 28, divisible by 4.
- n = 7 gives 49 + 3 = 52 = 4 × 13, which is divisible by 4. Four cases agree, but that is evidence, not proof.
2. Disprove: “n² > 3n for every whole number n ≥ 1.”
- Try small values first. n = 1 gives 1 and 3; 1 is not greater than 3.
- That one case is a counterexample, so the statement is false. (It also fails at n = 2 and n = 3, where n² = 4 and 9 against 6 and 9.)
3. Predict from a pattern: 1, 1 + 3 = 4, 1 + 3 + 5 = 9. What is the sum of the first 4 odd numbers?
- The sums 1, 4, 9 are the squares 1², 2², 3². The conjecture is: the sum of the first n odd numbers is n².
- For n = 4 it predicts 16, and 1 + 3 + 5 + 7 = 16. The check agrees, but it is still only a conjecture.
Your go. The first step is given. Disprove: “4n + 3 is prime for every whole number n ≥ 1.” Try n = 1, 2, 3 in order.
- n = 1 gives 4 + 3 = 7, which is prime. n = 2 gives 8 + 3 = 11, which is prime.
- Now try n = 3. If the value is not prime, write the counterexample as a sentence with the numbers.
Check the answer
n = 3 gives 4 × 3 + 3 = 15 = 3 × 5, which is not prime, so n = 3 is a counterexample and the statement is false. (Whenever n is a multiple of 3, 4n + 3 is a multiple of 3: n = 6 gives 27.)
With a GDC (optional)
A GDC can test many cases quickly. Enter the expression with the variable X, open the table of values and set the start and step to 1. Scroll down the list and look for a value that has a factor other than 1 and itself.
A table of values is evidence, not a proof: it can only show the cases it lists. Some GDCs will also list the factors of a number; if yours does not, divide by 2, 3, 5, 7 and 11 by hand. Everything here can be done by hand.
Check before you go on
1. A claim says: “every number ending in 3 is prime.” Which of these shows the claim is false?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. Kofi says: “6n + 1 is prime for every whole number n ≥ 1.” What is the smallest value of n that shows he is wrong?
3. A claim says: “n² ≥ 2n for every whole number n from 0 to 5.” Which value of n makes it false?
Level 2 · Building
4. Aiko says: “n² + n + 11 is prime for every whole number n ≥ 1.” What is the smallest value of n that shows she is wrong?
5. For how many whole numbers n from 1 to 15 is n² − n + 11 a prime number?
Level 3 · Stretch
6. A conjecture says: “if n is a prime number, then 2ⁿ − 1 is prime.” What is the smallest prime n that is a counterexample? (2ⁿ means 2 multiplied by itself n times.)
7. The first even number is 2, and the sum of the first 2 even numbers is 2 + 4 = 6, and of the first 3 is 2 + 4 + 6 = 12. The pattern continues for the first 4 as 20 and the first 5 as 30. Using the pattern, what is the sum of the first 10 even numbers?
Something to think about
A pattern gave the first odd number as 1, the sum of the first 2 odd numbers as 4 and of the first 3 as 9. If you test the first 20 and it still fits, have you proved it? What would a proof have to do that testing cannot?
One way to think about it
No. Twenty cases is still only twenty cases. A proof has to cover every n at once, which means working with a letter, as in the next lesson. Until then the pattern is a conjecture, a very well supported one.
Exit ticket
8. Zainab claims that n³ + n is even for every whole number n. She tests n = 1 to 9 and every value is even. Which statement is correct?