Smallest prime where the successor of the product of primes up to it is not prime

Find a prime number p such that the number (2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅⋅⋅ p) + 1 is not prime.

Let's break this problem down step by step.

Problem restatement:

Find a prime number $p$ such that the number

$$(2\cdot 3\cdot 5\cdot 7\cdots p)+1$$

is not prime.

This is the product of all primes up to $p$ (the primorial of $p$) plus 1. You're being asked for the smallest prime $p$ where this is not prime (i.e., composite).

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Step 1: Test Small Values of $p$

Let's compute $(2\cdot 3\cdot 5\cdots p)+1$ for successive primes until it's not prime.

1. $p=2$:

$$2+1=3\text{ (prime)}$$

2. $p=3$:

$$2\cdot 3+1=6+1=7\text{ (prime)}$$

3. $p=5$:

$$2\cdot 3\cdot 5+1=30+1=31\text{ (prime)}$$

4. $p=7$:

$$2\cdot 3\cdot 5\cdot 7+1=210+1=211\text{ (prime)}$$

5. $p=11$:

$$2\cdot 3\cdot 5\cdot 7\cdot 11+1=2310+1=2311\text{ (test primality)}$$

Test 2311:

• Not divisible by 13 (2311/13 = 177.76)
• Not divisible by 17, 19, ... (test shows 2311 is prime)

6. $p=13$:

$$2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13+1=30030+1=30031$$

Test 30031: Check divisibility:

• 30031 divided by 59:

$$30031/59=509$$

Check:

$$59\cdot 509=30031\text{ ✅ }$$

So 30031 is composite.

✅ Therefore, the smallest prime $p$ such that $(2\cdot 3\cdot 5\cdots p)+1$ is not prime is

$$\boxed{{\displaystyle 13}}$$