Let P(n) be the product of four consecutive integers, starting with n, minus one.

P(n) = n (n+1) (n+2) (n+3) - 1

Of course, P(n) can be prime or not. : P(1) = 23 is prime and P(2) = 119 = 7*17 is not.

I call " k-tuple " k consecutive values : { n, n+1, n+2, ..., n+k-1} such that : P(n), P(n+1), ... , P(n+k-1) are all prime.

There is a large number of twins (k = 2) or triplets (k = 3) but the quadruplets (k = 4) are rather sparse, not to speak about 5 or 6-tuples.

Here, for each k, I give the smallest value of n that I have found :

k = 2 n = 3 : P(3) = 359 and P(4) = 839 are both primes

k = 3 n = 6 : P(6), P(7), P(8) are primes (respectively 3023, 5039, 7919)

k = 4 n = 7046

2*3*4*5-1 = 119 = 0 mod 7, hence 7 | P(7n+2)

On my request, Gronau explained in non-modular terms:

If we add to every factor 7, we get:

(2+7)*(3+7)*(4+7)*(5+7) - 1.

This is

2*3*4*5 + 2*3*4*7 + 2*3*7*5 + ...(many other terms containing one or more 7's)... - 1

BUT: If we are only interested in the divisibility by 7, we can throw away any product containing 7. The only product containing no 7s is again: 2*3*4*5. So we have

P(2+7) = 2*3*4*5 + (something divisible by 7) - 1.

Because also 2*3*4*5-1 is divisible by 7, this is true for the whole thing. Of course nothing changes if you add not 7 but any multiple of 7. This means: 7 | P(7k + 2) for every k

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The most of all the puzzles got partial solutions for the question 2, but only J. K. Andersen & Igor Schein got them all:

For k=5: Start at n = 6985660, 22083548, 26203678, 30014408.

For k=6: Start at n = 512176360, 6958923639, 9427700628, 10205244472.

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I have not the details of the shortcuts used by Andersen, but other puzzlers sent to me  several hints of their respective approach:

From Igor:

First, I use the fact that P(n) is (((n+6)*n+11)*n+6)*n-1 - it's faster than dump multiplication. 2nd, I use recursion

P(n+1) = (P(n)+1)/n*(n+4)-1.

Now, in order to find a 6-tuple I need P(7*m-4),...,P(7*m+1) to be pseudoprime. So I test them in order until I hit a composite, in which case I jump straight to P(7*(m+1)-4). That saves unnecessary primality testing....

It's possible to improve further, using the fact that P(n) is the 1st prime of a 6-tuple only if n is 6, 7, 8 or 9 mod 23. Take other

primes like 17, 31 etc. and you'll get more congruence inequalities, reducing the total computational time...

Actually, P(n)=((n+3)*n+1)^2-2 is a more effective formula - only 5 arithmetic operations instead of 7. It doesn't make much difference in my approach, because I use recursion, which is also 5 operations, but if one chooses to sieve more aggressively, then this formula may come in handy.

From Ken Wilke:

I noticed the following observations which make programming easier.

P(n) = n(n+1)(n+2)(n+3) -1 = (n^2 + 3n +1)^2 -2. Clearly P(n) has the form 8m + 7 for some

integer m. Now if P(n) is prime, P(n) is a solution of the congruence x^2 = 2 (mod P(n)). This means that 2 is a quadratic residue (mod P(n)). Thus 2^((P(n)-1)/2) = 1((mod P(n)). Then

2^(1+(P(n)-1)/2) =2^((P(n)+1)/2) = 2 = x^2((mod P(n)) so x =

+2^((P(n)+1)/4) ((mod P(n)).

Thus x = 2^((P(n)+1)/4) or P(n) - 2^((P(n)+1)/4) (mod P(n)).

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Here are the details of the search by Andersen: