This is an account of the third session with L on proving the statement

*'All primes >3 are of the form 6n+-1'*. Since our last session he had created the following table:

He had highlighted the numbers 119 and 121 as examples that proved the statement to be

*false*. After a while, it became clear that he thought this because

*there was no prime one less*.

__or__one more than 120This is logically very tricky, I think. To disprove the statement we would need to find a prime that is not of the form 6n+-1, not a non-prime of the form 6n+-1. This would mean extending the search beyond the numbers shown.

We talked about this for a while, sometimes using the statement 'All cows have 4 legs', which was sometimes useful and sometimes not. At some point it became clear that what we had only checked 4-legged animals, and would need to check other-legged animals to disprove/prove the statement.

As we talked about the numbers we had checked, L said:

*"The numbers are in a funny order."*He moved the 6n-1 column to the left of the 6n column, and then spontaneously starting filling in the other numbers (8,9,10,14,15...), as shown below:

This was an important moment. L looked at the new numbers and said,

*"These are not primes!"*I replied:

*"How do you know?"*He started checking some of the new numbers for primeness:

L checked five numbers (8, 34, 22, 27, 39) for primeness. This checking is an important part of

*getting-a-sense-of*, but I felt an intervention was required in order to guide him towards the idea of

*looking at the whole*, rather than continuing to check individual examples. He noted that the columns contained evens and odds, and it was at this point that he entered the column headings shown above.

As he was about to check 45, I asked:

*"Can you make a conjecture about whether this number is prime?"*He said

*"It's not prime, because of 27 and 39."*We talked about how each number in the 'odds' column was divisible by 3, so there were no primes in any of these new numbers.

At this point, L asked:

*"So, where are we at?"*This is what I call '

*returning to the problem*'. I invited him to look at the statement, and look at what he had done. He felt it would

*simplify*everything to delete all references to cows, during which he spontaneously started

*relabelling*the even/odds columns with algebraic expressions 6n+2, 6n+3, ... as shown below:

This felt significant. After a valuable discussion about 6n+5 being equivalent to 6n-1 (with different values for n), he arrived at this:

At this stage, he asked a '

*what-if*' question:

*"What if the statement we were trying to prove was ... of the form 6n+-2?"*. He subsequently showed that this statement would be false. This is in some sense a red herring, but perhaps not; what-if questions are certainly something I would encourage a mathematician to create.

He then returned his attention to the example of 119 and 121 that we started with. At this point, I felt some guidance was required, in order to avoid going in a circle. I asked him to look in the 'primes' column (on the left) and invited him to

*"Find a prime that is not of the form 6n+-1"*. He could see that all of the primes 5 to 139 were in the two columns 6n +- 1, but what about some other prime? He tested the (known) prime 8293 and and found it was one more than 6 x 1382. He then tested 10169 and found it was one less than 6 x 1695.

Then something suddenly clicked. He exclaimed:

*"119 and 121 didn't matter!"*He could see that they did not constitute a counterexample to the statement. I asked him if he thought the statement was true. He said that he was convinced it was, mostly because of the two

*"random examples"*that he had just tested. Although these two examples do not make a proof, they were valuable in that they convinced him of the truth of the statement, and allowed him to move on from his fixation with 119 and 121 as counterexamples.

At this point, L pondered whether it would be fruitful to test the prime 56,597, but I think he also realised that this would not be of any use as a proof. I asked him what column he thought 56,597 would be in. He thought it would be in one of the two columns 6n+-1. I left him with the question:

*"How could you know that 56,597 - and indeed*

__any__

*prime -*

__must__

*be in one of these two columns?"*