Find the smallest prime number N such that the following is true:

The largest prime factor of N−1 is A;
The largest prime factor of A−1 is B;
The largest prime factor of B−1 is 7.

To find the smallest prime number N that satisfies the given conditions, we can follow these steps:

1. Start by finding the largest prime factor of 7. Since 7 is a prime number itself, it has no factors other than 1 and itself. Therefore, the largest prime factor of 7 is 7.

2. Next, let's find the largest prime factor of B-1. Since we don't have any information about B, we need to find the smallest prime number that has 7 as its largest prime factor. To do this, we can start with 7 and keep adding 7 until we find a prime number.

- 7 * 1 = 7 -> not a prime number
- 7 * 2 = 14 -> not a prime number
- 7 * 3 = 21 -> not a prime number
- 7 * 4 = 28 -> not a prime number
- 7 * 5 = 35 -> not a prime number
- 7 * 6 = 42 -> not a prime number
- 7 * 7 = 49 -> not a prime number
- 7 * 8 = 56 -> not a prime number
- 7 * 9 = 63 -> not a prime number
- 7 * 10 = 70 -> not a prime number
- ...

After trying several multiples of 7, we find that 7 * 11 = 77 is the smallest number that has 7 as its largest prime factor.

3. Let's continue by finding the largest prime factor of A-1. Again, we don't have any information about A, so we need to find the smallest prime number that has 77 as its largest prime factor. We'll follow the same process as before.

- 77 * 1 = 77 -> not a prime number
- 77 * 2 = 154 -> not a prime number
- 77 * 3 = 231 -> not a prime number
- 77 * 4 = 308 -> not a prime number
- 77 * 5 = 385 -> not a prime number
- 77 * 6 = 462 -> not a prime number
- 77 * 7 = 539 -> not a prime number
- 77 * 8 = 616 -> not a prime number
- 77 * 9 = 693 -> not a prime number
- 77 * 10 = 770 -> not a prime number
- ...

After trying several multiples of 77, we find that 77 * 11 = 847 is the smallest number that has 77 as its largest prime factor.

4. Finally, we need to find the smallest prime number N for which 847-1 = N-1. This equation can be simplified as N = 846 + 1.

The prime factorization of 846 is 2 * 3 * 3 * 47. To find the next prime number, we increment the value of N by 1 until we find a prime number.

- N = 846 + 1 = 847 -> not a prime number
- N = 847 + 1 = 848 -> not a prime number
- N = 848 + 1 = 849 -> not a prime number
- N = 849 + 1 = 850 -> not a prime number
- N = 850 + 1 = 851 -> not a prime number
- N = 851 + 1 = 852 -> not a prime number
- N = 852 + 1 = 853 -> prime number!

Therefore, the smallest prime number N that satisfies the given conditions is N = 853.