Mean of Consecutive Primes

This brain teaser asks about the behavior of primes when you average two that are adjacent in the prime sequence. The setup focuses on consecutive primes and whether taking their arithmetic mean can ever land you back on a prime. It probes how you think about "gaps" in a sequence and what it means, structurally, for two elements to be consecutive. Although the topic is elementary number theory, the framing is more about logical precision than heavy mathematics and is common in general brainteaser or puzzle-style interviews rather than specialist quant roles.

To answer well, a candidate needs to use basic properties of strictly increasing sequences and of averages, and connect those to the definition of consecutiveness. The interviewer is watching for a clean, rigorous argument rather than trial-and-error with specific primes. Strong answers formalize intuitions like "nothing lies between consecutive elements" and apply them to the prime setting, possibly generalizing the reasoning to other sequences. This tests clarity of thought, comfort with simple proof structures, and the ability to generalize from a concrete question.

What it tests

When dealing with sequences defined by gaps—such as consecutive primes—any value strictly between two consecutive elements cannot itself be an element of the sequence. This is because the definition of consecutiveness in such sequences means there are no other elements between the two. For primes, this means that if `p` and `q` are consecutive, any number strictly between `p` and `q` cannot be prime. The arithmetic mean of two distinct numbers always lies strictly between them, so for any two consecutive primes, their mean cannot be prime. This principle generalizes to any strictly increasing sequence where 'consecutive' means 'no other sequence elements in between.'