[mfoc]

When choosing rows = 4000 and columns = 546, an interesting pattern emerges.

For all integers n ≥ 0, the ranges [243 + (n * 546)] to [249 + (n * 546)] inclusive appear to contain no prime numbers. Same with the ranges [297 + (n * 546)] to [303 + (n * 546)].

For both sets of ranges, the minimum gap between the closest neighbouring primes appears to be at least 10 (in decimal). Does anyone know of a number-theoretic explanation for this kind of pattern?

[Karliss]

Nothing too surprising more or less same thing as all other vertical columns. 546=2*3*7*13 and all numbers in the range [242, 250] can be divided by one of those 4 primes same for [296, 304] . If x is divisor of a and b, then it will also divide a+n*b meaning you get an empty column. 2 and 3 already makes more than half the columns non primes, fill in the gaps with few more primes and you get the wide empty columns. If the width is multiple of many different small primes it's more likely to happen.

In a similar way 210=2*3*7*5 also gives wide empty columns (if you ignore ignore first row where 2,3,5,7 themselves are primes)

It helps if you think of it in terms of where the non primes are located instead of where the primes are. Multiples of 2, 3, 5 form a very regular pattern. Wrap it around in a grid and you get straight lines which are either straight vertical or slightly shifted depending on the divisors of width. Stack a couple of repetitive patterns and you still get a repetitive pattern. If the positions which are not primes form a regular pattern, the inverted image also forms recognizable pattern. Of course the primes don't form perfectly regular pattern and but most the visible repetition are result of small prime multiples.

[mfoc]

Thank you.

[AnotherGoodName]

Those are just columns that happen to share factors with 546 and happen to line up together. 546 is a very composite number (lots of factors!)

546n + 242 is always even

546n + 243 is always divisible by 3

546n + 244 is always even

546n + 255 is always divisible by 7.

Etc.

It's similar to how in base 10 you never see a prime ending in 0,2,4,6,8 or 5 since those numbers are clearly divisible by 2 or 5. In 546's case you have a lot of factors so even more gaps.

You also get the pattern appearing again since the number is even and the non-even factors repeat their pattern starting from the halfway point.

[mfoc]

That makes perfect sense. Thank you.

To finish it off (skipping even numbers)...

546n + 247 is always divisible by 13.

546n + 249 is always divisible by 3.

[voisin]

I am interested to know how you discovered this! Was it by happenstance or did you know to look for this?

[mfoc]

I had noticed it before. The response by AnotherGoodName makes it clear to me now why.