**Definition**: In the gap between any two consecutive odd primes we have one or more composite numbers. One of these composite number will have a prime factor which is greater than that of any other number in the gap. I am interested in this prime factor which is the largest in this gap.

> **Claim**: Every prime is the largest prime factor in some prime gap.

I am looking for a proof or disproof.

**Update, 7 Dec 2019**: Conjecture verified for $p \le 2 \times 10^9.$

**Note**: [This question was posted in MSE and got many upvotes but no answer hence posting in MO][1].


  [1]: https://math.stackexchange.com/questions/3456944/largest-prime-factor-of-the-numbers-between-two-consecutive-primes