Question 21 of 36intermediate🔧 ApplyNumerical3 marks

Observe that 3 is a prime number, and 2×32 \times 3 + 1 = 7 is also a prime. Are there other primes for which doubling and adding 1 gives another prime? Find at least five such examples.

Correct Answer

Five examples of primes p where 2p + 1 is also prime:

Step 1: Test p = 2: 2(2) + 1 = 5 (prime) ✓ Step 2: Test p = 3: 2(3) + 1 = 7 (prime) ✓ Step 3: Test p = 5: 2(5) + 1 = 11 (prime) ✓ Step 4: Test p = 11: 2(11) + 1 = 23 (prime) ✓ Step 5: Test p = 23: 2(23) + 1 = 47 (prime) ✓

Answer: Five such primes are p = 2, 3, 5, 11, and 23.

Exercise: Figure it Out (Section 3) | Q: 12 | (Page Page 9)
For More Understanding

Explanation

We need primes p such that 2p + 1 is also prime. Testing: p=2: 2(2)+1=5 (prime) ✓; p=3: 2(3)+1=7 (prime) ✓; p=5: 2(5)+1=11 (prime) ✓; p=7: 2(7)+1=15 (not prime); p=11: 2(11)+1=23 (prime) ✓; p=13: 2(13)+1=27 (not prime); p=17: 2(17)+1=35 (not prime); p=19: 2(19)+1=39 (not prime); p=23: 2(23)+1=47 (prime) ✓.

Solution Steps

  1. Step 1: Test p = 2: 2(2) + 1 = 5 (prime) ✓

  2. Step 2: Test p = 3: 2(3) + 1 = 7 (prime) ✓

  3. Step 3: Test p = 5: 2(5) + 1 = 11 (prime) ✓

  4. Step 4: Test p = 11: 2(11) + 1 = 23 (prime) ✓

  5. Step 5: Test p = 23: 2(23) + 1 = 47 (prime) ✓

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