One of the problems that has come up during my research concerns $K_4$-simple groups (simple groups with $4$ prime divisors). The only infinite family of groups satisfying this condition is $PSL(2,q)$, and I was interested in exactly what values of $q$ satisfied this condition. After writing a quick Python program, I got this output (where each ordered pair denotes $(q, |PSL(2,q)|)$:

$$[(11, 660), (13, 1092), (16, 4080), (19, 3420), (23, 6072), (25, 7800), (27, 9828), (31, 14880), (32, 32736), (37, 25308), (47, 51888), (49, 58800), (53, 74412), (73, 194472), (81, 265680), (97, 456288), (107, 612468), (127, 1024128), (128, 2097024), (163, 2165292), (193, 3594432), (243, 7174332), (257, 8487168), (383, 28090752), (487, 57750408), (577, 96049728), (863, 321367392), (1153, 766403712), (2187, 5230175508), (2593, 8717209632), (2917, 12410213148), (4373, 41812719372), (8192, 549755805696), (8747, 334616519988), (131072, 2251799813554176), ...$$

There seems to be a huge gap in $q$ after $8747$, going immediately to $2^{17} = 131072$ and then skipping another couple hundred thounsand or so. The condition on $q$ is that it is a prime power satisfying $q(q^2-1) = 2^{\alpha_1}3^{\alpha_2}p^{\alpha_3}r^{\alpha_4}$ for primes $r > p > 3$. Yet, I found it quite surprising that these gaps exist. Does anyone know why? Are there arbitrarily large gaps in values of $q$?

This is more number-theoretic than group-theoretic, but I am definitely interested in that aspect as well. Thanks in advance!

One of the problems that has come up during my research concerns $K_4$-simple groups (simple groups with $4$ prime divisors). The only (potentially) infinite family of groups satisfying this condition is $PSL(2,q)$, and I was interested in exactly what values of $q$ satisfied this condition. After writing a quick Python program, I got this output (where each ordered pair denotes $(q, |PSL(2,q)|)$:

$$[(11, 660), (13, 1092), (16, 4080), (19, 3420), (23, 6072), (25, 7800), (27, 9828), (31, 14880), (32, 32736), (37, 25308), (47, 51888), (49, 58800), (53, 74412), (73, 194472), (81, 265680), (97, 456288), (107, 612468), (127, 1024128), (128, 2097024), (163, 2165292), (193, 3594432), (243, 7174332), (257, 8487168), (383, 28090752), (487, 57750408), (577, 96049728), (863, 321367392), (1153, 766403712), (2187, 5230175508), (2593, 8717209632), (2917, 12410213148), (4373, 41812719372), (8192, 549755805696), (8747, 334616519988), (131072, 2251799813554176), ...$$

There seems to be a huge gap in $q$ after $8747$, going immediately to $2^{17} = 131072$ and then skipping another couple hundred thounsand or so. The condition on $q$ is that it is a prime power satisfying $q(q^2-1) = 2^{\alpha_1}3^{\alpha_2}p^{\alpha_3}r^{\alpha_4}$ for primes $r > p > 3$. Yet, I found it quite surprising that these gaps exist. Does anyone know why? Are there arbitrarily large gaps in values of $q$?

This is more number-theoretic than group-theoretic, but I am definitely interested in that aspect as well. Thanks in advance!

One of the problems that has come up during my research concerns $K_4$-simple groups (simple groups with $4$ prime divisors). The only infinite family of groups satisfying this condition is $PSL(2,q)$, and I was interested in exactly what values of $q$ satisfied this condition. After writing a quick Python program, I got this output (where each ordered pair denotes $(q, |PSL(2,q)|)$:

$$[(11, 660), (13, 1092), (16, 4080), (19, 3420), (23, 6072), (25, 7800), (27, 9828), (31, 14880), (32, 32736), (37, 25308), (47, 51888), (49, 58800), (53, 74412), (73, 194472), (81, 265680), (97, 456288), (107, 612468), (127, 1024128), (128, 2097024), (163, 2165292), (193, 3594432), (243, 7174332), (257, 8487168), (383, 28090752), (487, 57750408), (577, 96049728), (863, 321367392), (1153, 766403712), (2187, 5230175508), (2593, 8717209632), (2917, 12410213148), (4373, 41812719372), (8192, 549755805696), (8747, 334616519988), (131072, 2251799813554176), (416147, 36033820333396192), (416333, 36082158684990848), ...]$$

There seems to be a huge gap in $q$ after $8747$, going immediately to $2^{17} = 131072$ and then skipping another $300,000$ or so. The condition on $q$ is that it is a prime power satisfying $q(q^2-1) = 2^{\alpha_1}3^{\alpha_2}p^{\alpha_3}r^{\alpha_4}$ for primes $r > p > 3$. Yet, I found it quite surprising that these gaps exist. Does anyone know why? Are there arbitrarily large gaps in values of $q$?

This is more number-theoretic than group-theoretic, but I am definitely interested in that aspect as well. Thanks in advance!

One of the problems that has come up during my research concerns $K_4$-simple groups (simple groups with $4$ prime divisors). The only infinite family of groups satisfying this condition is $PSL(2,q)$, and I was interested in exactly what values of $q$ satisfied this condition. After writing a quick Python program, I got this output (where each ordered pair denotes $(q, |PSL(2,q)|)$:

$$[(11, 660), (13, 1092), (16, 4080), (19, 3420), (23, 6072), (25, 7800), (27, 9828), (31, 14880), (32, 32736), (37, 25308), (47, 51888), (49, 58800), (53, 74412), (73, 194472), (81, 265680), (97, 456288), (107, 612468), (127, 1024128), (128, 2097024), (163, 2165292), (193, 3594432), (243, 7174332), (257, 8487168), (383, 28090752), (487, 57750408), (577, 96049728), (863, 321367392), (1153, 766403712), (2187, 5230175508), (2593, 8717209632), (2917, 12410213148), (4373, 41812719372), (8192, 549755805696), (8747, 334616519988), (131072, 2251799813554176), ...$$

There seems to be a huge gap in $q$ after $8747$, going immediately to $2^{17} = 131072$ and then skipping another couple hundred thounsand or so. The condition on $q$ is that it is a prime power satisfying $q(q^2-1) = 2^{\alpha_1}3^{\alpha_2}p^{\alpha_3}r^{\alpha_4}$ for primes $r > p > 3$. Yet, I found it quite surprising that these gaps exist. Does anyone know why? Are there arbitrarily large gaps in values of $q$?

This is more number-theoretic than group-theoretic, but I am definitely interested in that aspect as well. Thanks in advance!