Observe that 3 is a prime number, and is also a prime. Are there other primes for which doubling and adding 1 gives another prime? Find at least five such examples.

Question

Homi · Accepted Answer

We will check small prime numbers to see if doubling them and adding one also gives a prime number.

Step 1 — Understand prime numbers

A prime number is a special number. It can only be divided by 1 and itself. For example, 2 is prime. 3 is prime. 4 is not prime because $4 = 2 \times 2$ . Let us list some small prime numbers. They are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, ...

Step 2 — Test prime numbers

We will take each prime number. We will multiply it by 2. Then we will add 1. We will check if the new number is prime.

Let us start with the smallest prime, 2. We take 2. $2 \times 2 + 1$ $= 4 + 1$

$\boxed{5}$ Is 5 a prime number? Yes, it is. So, 2 is our first example.

Next, let us try 3. The question already gave this example. $2 \times 3 + 1$ $= 6 + 1$

$\boxed{7}$ Is 7 a prime number? Yes, it is.

Now, let us try 5. We take 5. $2 \times 5 + 1$ $= 10 + 1$

$\boxed{11}$ Is 11 a prime number? Yes, it is. So, 5 is our second example.

Next, let us try 7. We take 7. $2 \times 7 + 1$ $= 14 + 1$

$\boxed{15}$ Is 15 a prime number? No, because $15 = 3 \times 5$ . So, 7 is not an example.

Next, let us try 11. We take 11. $2 \times 11 + 1$ $= 22 + 1$

$\boxed{23}$ Is 23 a prime number? Yes, it is. So, 11 is our third example.

Next, let us try 13. We take 13. $2 \times 13 + 1$ $= 26 + 1$

$\boxed{27}$ Is 27 a prime number? No, because $27 = 3 \times 9$ . So, 13 is not an example.

Next, let us try 17. We take 17. $2 \times 17 + 1$ $= 34 + 1$

$\boxed{35}$ Is 35 a prime number? No, because $35 = 5 \times 7$ . So, 17 is not an example.

Next, let us try 19. We take 19. $2 \times 19 + 1$ $= 38 + 1$

$\boxed{39}$ Is 39 a prime number? No, because $39 = 3 \times 13$ . So, 19 is not an example.

Next, let us try 23. We take 23. $2 \times 23 + 1$ $= 46 + 1$

$\boxed{47}$ Is 47 a prime number? Yes, it is. So, 23 is our fourth example.

Next, let us try 29. We take 29. $2 \times 29 + 1$ $= 58 + 1$

$\boxed{59}$ Is 59 a prime number? Yes, it is. So, 29 is our fifth example.

Next, let us try 31. We take 31. $2 \times 31 + 1$ $= 62 + 1$

$\boxed{63}$ Is 63 a prime number? No, because $63 = 7 \times 9$ . So, 31 is not an example.

Next, let us try 37. We take 37. $2 \times 37 + 1$ $= 74 + 1$

$\boxed{75}$ Is 75 a prime number? No, because $75 = 3 \times 25$ . So, 37 is not an example.

Next, let us try 41. We take 41. $2 \times 41 + 1$ $= 82 + 1$

$\boxed{83}$ Is 83 a prime number? Yes, it is. So, 41 is our sixth example.

We have found more than five examples.

Answer

Yes, here are examples:

(i) $2 \times 2 + 1 = 5$ (prime) (ii) $2 \times 5 + 1 = 11$ (prime) (iii) $2 \times 11 + 1 = 23$ (prime) (iv) $2 \times 23 + 1 = 47$ (prime) (v) $2 \times 29 + 1 = 59$ (prime) (vi) $2 \times 41 + 1 = 83$ (prime)

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