There are no known ‘neat’ expressions that generate only primes! Find counterexamples to the following claims.
- (i) All numbers of the form 4n2 + 1 are prime.
- (ii) All numbers of the form n2 + n + 11 are prime.
- (iii) All numbers of the form 4n + 3 are prime.
Step-by-step solution
Idea: A claim about all numbers of a form is false if just one value gives a composite number. Try n = 1, 2, 3, … in order and test each result for small prime factors.
(i) All numbers of the form 4n2 + 1 are prime.
- n = 1, 2, 3 give 5, 17, 37 — all prime. n = 4 gives 4 × 16 + 1 = 65.½ mark
- 65 = 5 × 13 is composite, so n = 4 is a counterexample.½ mark
(ii) All numbers of the form n2 + n + 11 are prime.
- n = 1 to 9 give 13, 17, 23, 31, 41, 53, 67, 83, 101 — all prime. n = 10 gives 100 + 10 + 11 = 121.½ mark
- 121 = 11 × 11 is composite, so n = 10 is a counterexample. (n = 11 also works: 143 = 11 × 13.)½ mark
(iii) All numbers of the form 4n + 3 are prime.
- n = 1, 2, 3 give 7, 19, 67 — all prime. n = 4 gives 256 + 3 = 259.½ mark
- 259 = 7 × 37 is composite, so n = 4 is a counterexample.½ mark
Check: 5 × 13 = 65, 11 × 11 = 121, 7 × 37 = 259 ✓. (If n = 0 is allowed, it gives quick counterexamples too: 4 × 0 + 1 = 1 is not prime, and 40 + 3 = 4 is not prime.)
Answer to write in the exam
(i)
n = 4: 4n2 + 1 = 4 × 16 + 1 = 65
65 = 5 × 13, composite
∴ n = 4 is a counterexample.
(ii)
n = 10: n2 + n + 11 = 100 + 10 + 11 = 121
121 = 11 × 11, composite
∴ n = 10 is a counterexample.
(iii)
n = 4: 4n + 3 = 256 + 3 = 259
259 = 7 × 37, composite
∴ n = 4 is a counterexample.
Common mistakes that cost marks
- Trying a few values, finding primes, and concluding the claim is true. A claim about all n needs every case; one failure is enough to reject it.
- Calling 121 prime. It is 11 × 11.
- Not testing 259 for divisibility by 7: 7 × 37 = 259.
How this can come in the exam
For which value of n is n2 − n + 41 NOT prime?
- n = 10
- n = 20
- n = 40
- n = 41
Show answer
(D) n = 41
n = 41: 412 − 41 + 41 = 412 = 1681, not prime. The others give 131, 421 and 1601, which are prime.
Find a counterexample to ‘All numbers of the form 6n + 1 are prime’.
Show answer
n = 1, 2, 3 give 7, 13, 19 (prime) (1 mark). n = 4 gives 25 = 5 × 5, not prime: a counterexample (1 mark).Try one yourself
Find the smallest positive n for which n2 + n + 17 is not prime.
Show answer
n = 16: 256 + 16 + 17 = 289 = 17 × 17. (n = 1 to 15 all give primes.)
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