Twin Primes
Twin primes are pairs of prime numbers that have a difference of exactly 2. They are the closest that two prime numbers (greater than 2) can be to each other.
In Class 5, students explore twin primes as part of their study of prime and composite numbers. Recognising twin primes sharpens the understanding of prime numbers and builds a foundation for more advanced number theory.
Prime numbers are spread across the number line in an irregular pattern. Sometimes two primes appear very close together — separated by just 2. These close pairs are called twin primes. For example, 11 and 13 are both prime and their difference is 13 − 11 = 2, so they form a twin prime pair.
Finding twin primes is a good exercise because it requires two skills: (1) testing whether a number is prime, and (2) checking whether its neighbour (two steps away) is also prime. This combines divisibility knowledge with systematic checking.
What is Twin Primes - Class 5 Maths (Factors and Multiples)?
Twin primes are a pair of prime numbers that differ by exactly 2.
If p and p + 2 are both prime, then (p, p + 2) is a twin prime pair.
First few twin prime pairs:
| Pair | Difference | Twin Primes? |
|---|---|---|
| (3, 5) | 5 − 3 = 2 | Yes |
| (5, 7) | 7 − 5 = 2 | Yes |
| (11, 13) | 13 − 11 = 2 | Yes |
| (17, 19) | 19 − 17 = 2 | Yes |
| (29, 31) | 31 − 29 = 2 | Yes |
| (41, 43) | 43 − 41 = 2 | Yes |
Note: (2, 3) is NOT a twin prime pair because 3 − 2 = 1, not 2. However, (2, 3) is the only pair of consecutive prime numbers.
Types and Properties
Understanding which pairs are and are not twin primes:
Twin prime pairs up to 100:
- (3, 5)
- (5, 7)
- (11, 13)
- (17, 19)
- (29, 31)
- (41, 43)
- (59, 61)
- (71, 73)
Common mistakes — these are NOT twin primes:
- (2, 3) — difference is 1, not 2
- (7, 11) — difference is 4
- (13, 17) — difference is 4
- (23, 29) — difference is 6
- (9, 11) — 9 is not prime (9 = 3 × 3)
Special note on 5: The number 5 appears in two twin prime pairs: (3, 5) and (5, 7). This is the only number that belongs to two twin prime pairs.
Solved Examples
Example 1: Example 1: Identifying a Twin Prime Pair
Problem: Is (11, 13) a twin prime pair?
Solution:
Step 1: Check if both numbers are prime.
11: Factors are 1 and 11 only — prime.
13: Factors are 1 and 13 only — prime.
Step 2: Check the difference: 13 − 11 = 2.
Answer: Yes, (11, 13) is a twin prime pair.
Example 2: Example 2: Not a Twin Prime Pair (Not Prime)
Problem: Is (15, 17) a twin prime pair?
Solution:
Step 1: Check if both numbers are prime.
15 = 3 × 5 — not prime.
Step 2: Since 15 is not prime, (15, 17) cannot be a twin prime pair.
Answer: No, (15, 17) is not a twin prime pair because 15 is composite.
Example 3: Example 3: Not a Twin Prime Pair (Wrong Difference)
Problem: Is (13, 17) a twin prime pair?
Solution:
Step 1: Both 13 and 17 are prime numbers.
Step 2: Difference = 17 − 13 = 4 (must be 2).
Answer: No, (13, 17) is not a twin prime pair because the difference is 4, not 2.
Example 4: Example 4: Finding Twin Primes in a Range
Problem: Find all twin prime pairs between 30 and 50.
Solution:
Step 1: List prime numbers between 30 and 50: 31, 37, 41, 43, 47.
Step 2: Check pairs with difference 2:
- 31 and 33: 33 = 3 × 11, not prime
- 37 and 39: 39 = 3 × 13, not prime
- 41 and 43: Both prime, difference = 2
- 47 and 49: 49 = 7 × 7, not prime
Answer: The only twin prime pair between 30 and 50 is (41, 43).
Example 5: Example 5: Twin Primes Between 50 and 80
Problem: Find all twin prime pairs between 50 and 80.
Solution:
Step 1: List prime numbers between 50 and 80: 53, 59, 61, 67, 71, 73, 79.
Step 2: Check pairs with difference 2:
- 53 and 55: 55 = 5 × 11, not prime
- 59 and 61: Both prime, difference = 2
- 67 and 69: 69 = 3 × 23, not prime
- 71 and 73: Both prime, difference = 2
- 79 and 81: 81 = 9 × 9, not prime
Answer: Twin prime pairs between 50 and 80: (59, 61) and (71, 73).
Example 6: Example 6: Why (2, 3) Is Not a Twin Prime
Problem: Kavi says (2, 3) is a twin prime pair because both are prime. Is he correct?
Solution:
Step 1: Both 2 and 3 are prime numbers.
Step 2: Difference = 3 − 2 = 1.
Step 3: Twin primes must have a difference of exactly 2, not 1.
Answer: No, Kavi is incorrect. (2, 3) is not a twin prime pair because the difference is 1. However, they are the only pair of consecutive primes.
Example 7: Example 7: The Number 5 in Two Pairs
Problem: Which number appears in two different twin prime pairs?
Solution:
Step 1: Consider the twin prime pairs: (3, 5) and (5, 7).
Step 2: The number 5 appears in both pairs.
Step 3: Can any other number appear in two twin prime pairs? For a number p to be in two pairs, both p − 2 and p + 2 must be prime. For p > 5, one of (p − 2, p, p + 2) is always divisible by 3.
Answer: 5 is the only number that belongs to two twin prime pairs.
Example 8: Example 8: Checking a Larger Pair
Problem: Is (101, 103) a twin prime pair?
Solution:
Step 1: Check if 101 is prime: not divisible by 2, 3, 5, or 7 (check up to square root of 101, which is about 10). 101 is prime.
Step 2: Check if 103 is prime: not divisible by 2, 3, 5, or 7. 103 is prime.
Step 3: Difference = 103 − 101 = 2.
Answer: Yes, (101, 103) is a twin prime pair.
Example 9: Example 9: Sum of a Twin Prime Pair
Problem: Observe the sum of each twin prime pair: (3,5), (5,7), (11,13), (17,19). What pattern do you see?
Solution:
Step 1: Calculate sums:
- 3 + 5 = 8
- 5 + 7 = 12
- 11 + 13 = 24
- 17 + 19 = 36
Step 2: Check: 12, 24, 36 are all divisible by 12.
Step 3: The sum of every twin prime pair (except the pair containing 3 and 5) is always divisible by 12.
Answer: The sum of every twin prime pair (after the first) is always divisible by 12.
Real-World Applications
Why study twin primes?
- Strengthens prime number understanding: Finding twin primes requires testing primality carefully.
- Number patterns: Observing twin primes reveals patterns in how primes are distributed.
- Mental maths practice: Checking for twin primes involves divisibility tests and factor finding — good mental exercise.
- Foundation for advanced maths: The Twin Prime Conjecture (whether there are infinitely many twin primes) is one of the oldest unsolved problems in mathematics.
Key Points to Remember
- Twin primes are pairs of prime numbers with a difference of exactly 2.
- First few twin prime pairs: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43).
- (2, 3) is NOT a twin prime pair — the difference is 1, not 2.
- 5 is the only number that appears in two twin prime pairs: (3, 5) and (5, 7).
- To check for twin primes: first verify both numbers are prime, then check the difference is 2.
- There are 8 twin prime pairs between 1 and 100.
- The sum of every twin prime pair (except the first) is divisible by 12.
- Both numbers in a twin prime pair (except 3 and 5) end in digits that sum to an even number.
Practice Problems
- List all twin prime pairs between 1 and 50.
- Is (27, 29) a twin prime pair? Explain why or why not.
- Find the twin prime pair between 80 and 100.
- Meera says (23, 25) is a twin prime pair. Is she correct? Give a reason.
- Find the sum of the twin prime pair (41, 43). Is this sum divisible by 12?
- Why can two even numbers never form a twin prime pair?
- What is the next twin prime pair after (71, 73)?
- Dev claims that every odd number between two twin primes is composite. Check this for the pair (11, 13) and (17, 19).
Frequently Asked Questions
Q1. What are twin primes?
Twin primes are pairs of prime numbers that differ by exactly 2. Examples include (3, 5), (11, 13), and (29, 31). Both numbers in the pair must be prime.
Q2. Is (2, 3) a twin prime pair?
No. Although both 2 and 3 are prime, their difference is 1, not 2. Twin primes must have a difference of exactly 2.
Q3. How many twin prime pairs are there between 1 and 100?
There are 8 twin prime pairs between 1 and 100: (3,5), (5,7), (11,13), (17,19), (29,31), (41,43), (59,61), and (71,73).
Q4. Can a composite number be part of a twin prime pair?
No. Both numbers in a twin prime pair must be prime. If either number is composite, the pair is not a twin prime pair.
Q5. What is the difference between co-prime numbers and twin primes?
Co-prime numbers are any two numbers whose HCF is 1 — they need not be prime. Twin primes are specifically pairs of prime numbers with a difference of 2. For example, (8, 15) are co-prime but not twin primes.
Q6. Do twin primes go on forever?
Mathematicians believe there are infinitely many twin prime pairs, but this has never been proved. It remains one of the oldest unsolved problems in mathematics, called the Twin Prime Conjecture.
Q7. Why does 5 appear in two twin prime pairs?
Because both 3 (which is 5 − 2) and 7 (which is 5 + 2) are prime. So 5 fits in both (3, 5) and (5, 7). For any number p greater than 5, one of p − 2, p, or p + 2 is always divisible by 3, making it impossible to belong to two pairs.
Q8. Are twin primes covered in NCERT Class 5?
Yes. Twin primes are introduced as part of the prime numbers and factors topic in the NCERT/CBSE Class 5 Mathematics curriculum.
