28277: A Prime To Be Proud Of

Until today my diurnal age has suffered a drought of prime numbers. Prior to today (3rd September 2026) when I turned 28277 days old, the last prime (28229) occurred on July 17th. This marked a gap of 48 days between successive primes. This is not a record gap but it is impressive. Figure 1 shows the successive record gaps between primes.

Figure 1

I'll enumerate some of 28277's most interesting properties:

PROPERTY 1:

It forms a twin prime with 28279 but it also marks the beginning of gaps of 2, 4, 6, 8 and 10 between successive primes. The progression of primes is thus 28277, 28279, 28283, 28289, 28297, 28307. Such an occurrence is not common and membership is restricted to only three numbers (13901, 21557, 28277) in the range up to 40000. These and subsequent numbers constitute OEIS A190817.

PROPERTY 2

28277 is what is called a "good" prime and I posted about this type of prime in my blog post titled The Good Prime on the 12th of March 2025. As I explained there:

A prime \(p_n\) is said to be \( \textbf{good} \) if \(p_n^{^\textbf{2}}>p_{n-i } \cdot p_{n+i} \) for all \( 1 \leq i < n \).

The good primes from 28277 to 40000 are: 

28277, 28387, 28403, 28493, 28537, 28571, 28591, 28597, 29833, 29983, 30011, 30059, 30089, 30491, 30631, 30637, 30671, 30757, 30803, 31121, 31139, 31147, 31957, 32027, 32051, 32057, 32297, 32969, 33287, 33311, 33329, 34123, 35729, 35747, 35797, 35801, 35831, 35951, 35963, 36433, 36451, 36467, 36523, 36527, 36671, 38113, 38149, 38167, 38177, 38543, 38557, 38593, 38651, 38669, 39079, 39089

PROPERTY 3

28277 has what might be called an "internal prime". Strip away the first and last digits and what remains is 827, a prime number. I discuss these types of numbers in my post titled Numbers Within Numbers from the 21st June 2026. Primes with this property form OEIS A069686:

 A069686: primes whose internal digits form a prime.

From 28277 to 40000, the members of the sequence are:

28277, 28279, 28297, 28393, 28537, 28571, 28573, 28579, 28591, 28597, 28631, 28771, 28813, 28817, 28837, 28871, 28879, 29077, 29191, 29297, 29411, 29473, 29531, 29537, 29671, 29717, 29833, 29837, 29917, 30029, 30059, 30071, 30113, 30119, 30133, 30137, 30139, 30197, 30293, 30313, 30319, 30431, 30539, 30593, 30671, 30677, 30713, 30839, 30893, 30971, 30977, 31013, 31019, 31033, 31039, 31079, 31091, 31139, 31271, 31277, 31319, 31379, 31391, 31393, 31397, 31511, 31513, 31517, 31573, 31793, 31799, 31817, 31973, 31991, 32117, 32119, 32233, 32237, 32297, 32299, 32411, 32413, 32573, 32579, 32633, 32693, 32713, 32717, 32719, 32771, 32779, 32831, 32833, 32839, 32933, 32939, 33071, 33073, 33113, 33119, 33179, 33311, 33317, 33377, 33479, 33493, 33533, 33599, 33679, 33739, 33791, 33797, 33893, 34019, 34211, 34213, 34217, 34313, 34319, 34337, 34439, 34499, 34613, 34631, 34673, 34679, 34871, 34877, 34913, 34919, 35099, 35419, 35573, 35771, 35879, 35933, 35993, 35999, 36011, 36013, 36017, 36073, 36131, 36137, 36191, 36313, 36319, 36433, 36473, 36479, 36599, 36739, 36779, 36833, 36913, 36919, 37013, 37019, 37097, 37199, 37273, 37277, 37337, 37339, 37397, 37511, 37517, 37571, 37573, 37579, 37619, 37691, 37693, 37699, 37871, 37879, 38113, 38119, 38219, 38231, 38237, 38239, 38273, 38299, 38393, 38593, 38639, 38833, 38839, 38873, 39079, 39113, 39119, 39191, 39199, 39293, 39371, 39373, 39419, 39671, 39679, 39719, 39779, 39839, 39971, 39979

PROPERTY 4

28277 gives prime 31931717 when digits become indices of prime numbers. Here we have:

• \(2 \rightarrow p_2=3\)
• \(8 \rightarrow p_8=19\)
• \(7 \rightarrow p_7=17\)

The primes of this sort from 28277 to 40000 are (permalink):

28277, 28289, 28319, 28429, 28807, 28979, 29137, 29347, 29399, 29717, 29819, 29837, 29917, 30059, 30089, 30169, 30187, 30367, 30467, 30469, 30497, 30509, 30689, 30697, 30707, 30727, 31139, 31159, 31177, 31247, 31277, 31337, 31469, 31489, 31687, 31847, 31849, 32099, 32309, 32569, 32717, 32957, 32987, 33049, 33469, 33577, 33629, 33797, 33809, 33889, 33937, 34019, 34129, 34259, 34297, 34327, 34439, 34469, 34607, 34649, 34807, 34819, 35027, 35159, 35407, 35597, 36037, 36107, 36109, 36467, 36469, 36587, 36637, 36809, 36857, 37087, 37139, 37199, 37277, 37337, 37579, 37889, 38189, 38299, 38327, 38459, 38609, 38639, 38699, 38747, 39047, 39119, 39239, 39397, 39667, 39779

PROPERTY 5

28277 is the average of a prime and its emirp in two different ways. The two ways are:

1. \( \dfrac{18773 + 37781}{2}= 28277\)
2. \( \dfrac{19763 + 36791}{2} = 28277\)

I discuss these sorts of primes in my blog post Prime Emirp Pair Averages of 12th May 2023. Such primes form OEIS A178587:

A178587

Primes that are the average of the members of more than one emirp pair.

These primes are few and far between with the initial members being:

14741, 22727, 23327, 24547, 25447, 27067, 28277, 42929, 63541, 65761, 85453, 1217171, 1221221, 1227271, 1243421, 1245421, 1246471, 1250521, 1253521, 1257521, 1261571, 1271671, 1283771, 1327231, 1335331, 1338331, 1339381 

PROPERTY 6

28277 has the following trajectory under the Primes(+) and Non_Primes(-) algorithm: 

\(28277 \rightarrow 28287 \rightarrow 28282 \rightarrow 28272 \rightarrow 28277\)

It is thus a vortical and part of the vortex beginning and ending with 28277. This vortex has 36 captives.

Numbers Within Numbers

Let's formalise the concept of internal digits. Take a number like 28211. It's prime but if we remove the leftmost digits (2) and the rightmost digit (1), we are left with the number 821. This is the number within a number and it happens to be prime also. It is thus a member of OEIS A069686:

 A069686: primes whose internal digits form a prime.

The members of this sequence from 28000 to 40000 are (permalink):

28097, 28099, 28111, 28211, 28219, 28277, 28279, 28297, 28393, 28537, 28571, 28573, 28579, 28591, 28597, 28631, 28771, 28813, 28817, 28837, 28871, 28879, 29077, 29191, 29297, 29411, 29473, 29531, 29537, 29671, 29717, 29833, 29837, 29917, 30029, 30059, 30071, 30113, 30119, 30133, 30137, 30139, 30197, 30293, 30313, 30319, 30431, 30539, 30593, 30671, 30677, 30713, 30839, 30893, 30971, 30977, 31013, 31019, 31033, 31039, 31079, 31091, 31139, 31271, 31277, 31319, 31379, 31391, 31393, 31397, 31511, 31513, 31517, 31573, 31793, 31799, 31817, 31973, 31991, 32117, 32119, 32233, 32237, 32297, 32299, 32411, 32413, 32573, 32579, 32633, 32693, 32713, 32717, 32719, 32771, 32779, 32831, 32833, 32839, 32933, 32939, 33071, 33073, 33113, 33119, 33179, 33311, 33317, 33377, 33479, 33493, 33533, 33599, 33679, 33739, 33791, 33797, 33893, 34019, 34211, 34213, 34217, 34313, 34319, 34337, 34439, 34499, 34613, 34631, 34673, 34679, 34871, 34877, 34913, 34919, 35099, 35419, 35573, 35771, 35879, 35933, 35993, 35999, 36011, 36013, 36017, 36073, 36131, 36137, 36191, 36313, 36319, 36433, 36473, 36479, 36599, 36739, 36779, 36833, 36913, 36919, 37013, 37019, 37097, 37199, 37273, 37277, 37337, 37339, 37397, 37511, 37517, 37571, 37573, 37579, 37619, 37691, 37693, 37699, 37871, 37879, 38113, 38119, 38219, 38231, 38237, 38239, 38273, 38299, 38393, 38593, 38639, 38833, 38839, 38873, 39079, 39113, 39119, 39191, 39199, 39293, 39371, 39373, 39419, 39671, 39679, 39719, 39779, 39839, 39971, 39979

We can modify the algorithm to search for square numbers within square numbers. For example, \(144 = 12^2\) and its internal digit \(4 = 2^2\) and so it is a member of OEIS A069701:

 A069701: squares with internal digits also forming a square > 0.

The members up to 40000 are few and far between: 

144, 196, 441, 841, 1369, 3364, 4096, 5041, 8649, 10816, 11449, 20164, 38416.

So the takeaway from this is that every \(n\) digit number has an (\(n-2\)) digit number inside of it provided \(n>2\) and sequences can be developed by relating this internal number to the larger number of which it forms a part.

Let's consider numbers that are biprime and whose internal digits also form a biprime. Take 28189 as an example:$$ \begin{align} 28189 &= 7 \times 4027 \\ 818 &= 2 \times 409 \end{align}$$These sorts of numbers are not listed in the OEIS. 

Similarly for triprimes. Take 28055 as an example:$$ \begin{align} 28055 &= 5 \times 31 \times  181\\805 &= 5 \times 7 \times 23 \end{align}$$Of course we could consider numbers within numbers within numbers etc. but given that I'm mainly examining five digit numbers less than 40000, I'll leave off at numbers within numbers. Of course, these relationships between numbers and their respective internal numbers only apply with certainty in the number base under consideration (base 10 so far in this post). Take our earlier example of 28211 and its internal number of 821. Both are prime. Let's change to base 9:$$ \begin{align} 28211_{10} &= 42625_9 \rightarrow 265_9 \text{ as its internal number} \\ 265_9 &= 218_{10} \text{ which is clearly not prime} \end{align} $$Primeness is base independent and if a number is not prime in a certain base then it is not prime in any base. However, the inner number while not prime in base 10 may be prime if regarded as a number in another base. Let's illustrate this with 28019. It is a prime in base 10 where the inner number, 801, is divisible by 3. However, we can note the following:$$ \begin{align} 801_{12} &= 1153_{10} \text{ which is prime}\\801_{15} &= 1801_{10} \text{ which is prime} \end{align} $$I've incorporated the search for the following types of numbers into my number analysis algorithm so that the following types of numbers can be identified:

• prime number whose internal number is also prime in any base from 2 to 16
• biprime number whose internal number is also prime using base 10 only
• triprime number whose internal number is also triprime using base 10 only
• square number whose internal number is also square using base 10 only