Here is the sequence of the first ten prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Do you see any pattern in this sequence? Can you think of a rule that can predict the next few prime numbers?
Answer: There is no regular pattern: the gaps are 1, 2, 2, 4, 2, 4, 2, 4, 6. Some facts hold (after 2 every prime is odd; after 3 every prime is 1 more or 1 less than a multiple of 6), but no simple rule gives the next prime. We find them by testing: the next primes are 31, 37, 41, 43, 47.
Step-by-step solution
Idea: Look at the differences. If they are not fixed and do not follow a pattern, there is no simple explicit or recursive rule; we must test each number for factors.
- Differences between neighbours: 1, 2, 2, 4, 2, 4, 2, 4, 6. They do not stay the same or grow in a regular way.½ mark
- Partial patterns: after 2 every prime is odd (an even number above 2 has the factor 2). After 3, each prime is next to a multiple of 6: 5, 7 (6 ± 1), 11, 13 (12 ± 1), 17, 19, 23, 29.1 mark
- But this is not a rule that predicts primes: 25 = 24 + 1 and 35 = 36 − 1 are next to multiples of 6, yet 25 = 5 × 5 and 35 = 5 × 7 are not prime.½ mark
- So we test the numbers after 29 for factors: 31 (prime), 33 = 3 × 11, 35 = 5 × 7, 37 (prime), 39 = 3 × 13, 41, 43 (prime), 45 = 5 × 9, 47 (prime). Next primes: 31, 37, 41, 43, 47. No simple formula for primes is known.1 mark
The primes show no regular pattern and no simple rule predicts them; testing for factors gives the next primes 31, 37, 41, 43, 47.
Answer to write in the exam
Differences: 1, 2, 2, 4, 2, 4, 2, 4, 6 (not regular)
After 2 all primes are odd; after 3 each is 6k ± 1, but 25 and 35 are of this form and not prime.
∴ No simple rule predicts the primes.
Testing 31 to 47 for factors: next primes are 31, 37, 41, 43, 47.
Common mistakes that cost marks
- Thinking every odd number is prime and predicting 31, 33, 35. 33 and 35 have factors (3 × 11, 5 × 7).
- Using the pattern of gaps (2, 4, 2, 4, …) as a rule. The gaps break the pattern (the gap after 23 is 6).
- Counting 1 as a prime. A prime has exactly two factors; 1 has only one.
How this can come in the exam
MCQ (1 mark)
Which of these numbers, each one more or one less than a multiple of 6, is NOT prime?
- 41
- 43
- 49
- 53
Show answer
(C) 49
49 = 48 + 1 but 49 = 7 × 7, so it is not prime.
Try one yourself
List the prime numbers between 50 and 70.
Show answer
53, 59, 61, 67 (51 = 3 × 17, 57 = 3 × 19, 63 = 7 × 9, 65 = 5 × 13, 69 = 3 × 23).
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