# Add new comment

Weird and wonderful things can happen when you set a ball in motion on a billiard table — and the theory of mathematical billiards has recently seen a breakthrough.

Was vaccinating vulnerable people first a good choice? Hindsight allows us to assess this question.

A game you're almost certain to lose...

What are the challenges of communicating from the frontiers of mathematical research, and why should we be doing it?

Celebrate Pi Day with the stars of our podcast,

*Maths on the move*!

Has anybody ever seen mentions of the following conjectures, very similar in appearance to Goldbach's or Lemoine's?

Lemoine’s conjecture is ”All odd integers greater than 5 can be represented as the sum of an odd prime number and an even semiprime”

We know that it can be can be rewritten:

2n + 1 = p + 2q always has a solution in primes p and q (not necessarily distinct) for n > 2.

It appears that the following very similar conjectures (also applying to the relations between an odd number, a prime and a semi-prime) can also be verified for values greater than 2 millions.

Additional conjecture 1: O = 2n + 1 = p - 2q always has a solution in distinct odd primes p and q for o >= 5

Additional conjecture 2: O = 2n + 1 = 2q - p always has a solution in p and q with q < p, p being an odd prime and q being 1 or an odd prime and o >= 1 (For every odd positive integer O there exists two distinct odd elements from PRIMES&1 p0 and p1 such that p1 is the average of O and p0 )

In addition, the Lemoine conjecture can be split into two different conjectures:

Additional conjecture 3: 2n + 1 = p + 2q always has a solution in p and q where p > q, p odd prime and q in PRIMES&1 for n > 2

And

Additional conjecture 4: 2n + 1 = p + 2q always has a solution in p and q where p <= q , q odd prime and p in PRIMES&1 for n > 2

Similarly the Goldbach conjecture:

E = 2n = p + q with E > 2 has a solution in primes p and q

Has a twin:

E = 2n = p – q with E > 2 has a solution in primes p and q

… but this conjecture, I believe, follows from the Goldbach conjecture and the fact that p+q which is even can always be rewritten as p-q