Where there cannot be the maximum number of primes

In a few posts from last year, I discussed pentatrigesimal representation of numbers using all numbers and all letters except O (“O” being a common reading of the number “0” and very similar in shape).

One interesting discovery I made is that whilst 35 consecutive numbers can contain as many as ten primes, if we tabulate divisibility of numbers between each multiple of 35, we find something quite interesting:

k	6n	6n+1	6n+2	6n+3	6n+4	6n+5
1		2		2	3	2
2	2		2		2	3
3	3	2		2		2
4	2	3	2		2
5	5	2	3	2	5	2
6	2		2	3	2
7	7	2	7	2	3	2
8	2		2		2	3
9	3	2		2		2
A	2	3	2	5	2	5
B		2	3	2		2
C	2		2	3	2
D		2		2	3	2
E	2	7	2	7	2	3
F	3	2	5	2	5	2
G	2	3	2		2
H		2	3	2		2
I	2		2	3	2
J		2		2	3	2
K	2	5	2	5	2	3
L	3	2	7	2	7	2
M	2	3	2		2
N		2	3	2		2
P	2		2	3	2
Q	5	2	5	2	3	2
R	2		2		2	3
S	3	2		2		2
T	2	3	2	7	2	7
U		2	3	2		2
V	2	5	2	3	2	5
W		2		2	3	2
X	2		2		2	3
Y	3	2		2		2
Z	2	3	2		2
Possible primes	8	8	8	8	8	8

What we see is that whilst there can be ten primes in a span of thirty-five numbers, there can as shown above never be more than eight primes between consecutive multiples of 35. The first occurrence of eight primes between multiples of 35 is between the pentatrigesimal numbers 4L0 and 4LZ, or in decimal 5,635 to 5,669, whereby 5,639, 5,641, 5,647, 5,651, 5,653, 5,657, 5,659 and 5,669 are all primes.

What I found reading today is that OEIS has actually tabulated the number of possible primes between multiples of base b, as well as the number possible in an arbitrary sequence of length n. I have compared “decades” and “centuries” in all bases up to 35, with sequences serving “centuries” (k00) shaded in grey:

Base		Arbitrary	Multiple	Difference
2		1	1	0
3		2	1	-1
2	4	2	2	0
5		2	2	0
6		2	2	0
7		3	2	-1
8		3	3	0
3	9	4	3	-1
10		4	4	0
11		4	4	0
12		4	4	0
13		5	4	-1
14		5	4	-1
15		5	4	-1
4	16	5	5	0
17		6	5	-1
18		6	6	0
19		6	6	0
20		6	6	0
21		7	5	-2
22		7	7	0
23		7	7	0
24		7	6	-1
5	25	7	7	0
26		7	7	0
27		8	7	-1
28		8	7	-1
29		8	8	0
30		8	7	-1
31		9	8	-1
32		9	9	0
33		10	8	-2
34		10	10	0
35		10	8	-2
6	36	10	10	0
7	49	13	12	-1
8	64	16	16	0
9	81	20	19	-1
10	100	23	23	0
11	121	27	26	-1
12	144	31	30	-1
13	169	37	36	-1
14	196	41	41	0
15	225	46	45	-1
17	289	58	56	-2
18	324	62	62	0
19	361	68	67	-1
20	400	75	75	0
21	441	81	80	-1
22	484	88	87	-1
23	529	95	94	-1
24	576	101	101	0
25	625	109	109	0
26	676	117	116	-1
27	729	124	124	0
28	784	132	131	-1
29	841	141	139	-2
30	900	149	149	0
31	961	158	156	-2

As you can see, the pentatrigesimal case I discovered earlier is not exceptional. For base 21, one already sees that there can never be more than five primes between multiples of 21, but there can be seven primes within arbitrary sequences of 21 numbers.

Nonetheless, the list is revealing because — as it did for me — it gave a false impression that there must be more prime-dense “decades” in pentatrigesimal than that from 4L0 to 4LZ (5,635 to 5,669 in decimal). In cases like this one does need to take some care looking for prime constellations that can actually be easily proved impossible.