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.

A Special Class of Twin Primes

I was born in the year 1949 and am very aware of the fact that this number is prime and that it forms a twin prime pair with 1951. Today I turned 27751 days old and noticed that this number and 27749 also form a pair of twin primes. I asked myself the question: how many twin prime pairs are there in the range up to 40000 that end in the digits 49 and 51.

This is an easy question to answer. The result appears below (permalink):

• 149 and 151
• 1049 and 1051
• 1949 and 1951
• 2549 and 2551
• 4049 and 4051
• 4649 and 4651
• 5849 and 5851
• 6449 and 6451
• 7349 and 7351
• 7949 and 7951
• 11549 and 11551
• 14249 and 14251
• 14549 and 14551
• 16649 and 16651
• 20549 and 20551
• 26249 and 26251
• 27749 and 27751
• 28349 and 28351
• 33149 and 33151
• 33749 and 33751
• 34649 and 34651

There are 21 pairs and these are:

(149, 151), (1049, 1051), (1949, 1951), (2549, 2551), (4049, 4051), (4649, 4651), (5849, 5851), (6449, 6451), (7349, 7351), (7949, 7951), (11549, 11551), (14249, 14251), (14549, 14551), (16649, 16651), (20549, 20551), (26249, 26251), (27749, 27751), (28349, 28351), (33149, 33151), (33749, 33751), (34649, 34651)

Plotted, the point pairs appear as shown in Figure 1, forming a perfectly straight line:

Figure 1: permalink

It's interesting to note that even when the 49 and 51 pairs are not both prime, they always seem to be \( \textbf{relatively prime} \) just as the initial numbers, 49 and 51, are. My conjecture is that adding an equal number of additional digits to the left of 49 and 51 does not change this relative primeness. In other words:$$ gcd( \dots \text{xxx}49 \text, \dots \text{xxx}51)=1$$These 49 and 51 number pairs will always be interesting because they surround the midpoint of centuries as reckoned by the span from one 0 to the next. Thus the midpoint of the numbers from 0 to 100 is 50, the midpoint of the numbers from 100 to 200 is 150 etc.

The algorithm is easily adapted to search for other prime pair digit endings but 49 and 51 are the ones that attract my interest. If we extend our search further we can find some interesting special cases. For example, there are pairs of primes that both start and end in 49 and 51 (the pairs can longer be twin primes of course). Up to one million, there are only four pairs and these are (permalink):

• 49549 and 51551
• 491149 and 511151
• 494749 and 514751
• 499549 and 519551

Read about the unrelated and largely non-mathematical \( \textbf{49 - 51 principle} \) (link). Here is a summary of the report (link) that Gemini Deep Research prepared to the prompt: 

What is significant, mathematically and otherwise, about the 51 : 49 ratio. Can you create a report that highlights the most interesting information associated with this important ratio?

The 51:49 ratio, while mathematically representing a near-even split with decimal equivalents of 0.51 and 0.49 (or 51% and 49%), holds a significance that extends far beyond its basic numerical properties. Its proximity to perfect equality often creates an initial perception of balance, yet this subtle deviation carries substantial weight in numerous real-world contexts. In demographics, it appears as a natural tendency in human birth rates. In voting, it frequently marks the threshold of a narrow but often decisive majority. In business, it defines power dynamics in equity partnerships and serves as the foundation for a cultural principle promoting generosity. Even in seemingly random events like a coin toss, a slight 51:49 bias has been observed.

The power of this slight imbalance is evident in competitive scenarios where it often dictates victory and control. In business, it highlights the delicate interplay between majority rule and the rights of the minority. Psychologically, a 51:49 split is perceived as close and can influence the emotional responses to wins and losses, as well as the sociological dynamics of near-even divisions within society. While not uniquely tied to major historical events in its precise form, the concept of a narrow majority it represents has been historically significant. Moreover, the "51/49 Principle" has emerged as a contemporary cultural phenomenon. Comparisons with other near-even ratios like 50.5:49.5 and 52:48 further underscore the subtle but important nuances associated with small numerical differences around the midpoint. Existing research across various fields confirms that the 51:49 ratio is not just a theoretical concept but a subject of empirical study with real-world implications.

In conclusion, the 51:49 ratio, though seemingly representing a minimal imbalance, often acts as a critical threshold or a subtle but important bias with significant consequences across a diverse range of fields. Its significance lies not just in its mathematical representation but in its ability to define outcomes, shape relationships, and influence perceptions in the complex tapestry of the real world. 

An Interesting Prime

Being born on the 3rd April 1949, my date of birth is often represented as 3 - 4 - 49. These numbers when concatenated form the prime number 3449. I was reminded of this number because of the factorisation of the number associated with my diurnal age today, 27592.$$27592 = 2^3 \times 3449 = 8 \times 3449$$So today my life can be divided into exactly eight equal parts, each of them 3449 days long which is about 9.44 years. The previous multiple$$7 \times 3449 = 24143$$occurred on May 10th 2015 when I was still working at the Shanghai Singapore International School. The next multiple$$9 \times 3449 = 31041$$ will fall on March 29th 2034, shortly before my 84th birthday (if I make it that far).

3449 forms the initial prime of a Cunningham chain of the first type with length exactly 3 and so: $$ 3449 \rightarrow 2 \times 3449 + 1 = 6889 \text{ (prime)} \\ 6889 \rightarrow 2 \times 6889 +1 = 13799 \text{ (prime)}$$Primes with this property form OEIS A059762. Another prime-related property of 3449 qualifies it for membership in OEIS A088483:

A088483: primes \( \textit{p} \) such that \(p^2+p-1\) and \(p^2+p+1\) are twin primes.

For 3449, the twin primes are \(11899049\) and \(11899051\). 

3449 is also a home prime with a homeliness of 3 because:$$ \begin{align} 611 &= 13 \times 47 \rightarrow 1347\\ 1347 &= 3 \times 449 \rightarrow 3449 \end{align}$$Not all primes are home primes of course. Take 613 as an example of a prime that is not a home prime because it cannot be formed by the concatenation of the prime factors of any number (the prime factors need to be concatenated in ascending order).

3449 is also a member of OEIS A153116:

A153116: primes \(p\) such that \(p^2 +12\) and \(p^2-12\) are also primes.

Here the two primes are \(11895589\) and \(11895613\). Additionally:$$ \text{period of}\frac{1}{3449}=\frac{3449-1}{8} = 431$$This property qualifies 3449 for membership of OEIS A056213:

A056213: primes \(p\) for which the period of reciprocal = \(\dfrac{p-1}{8}\).

3449 is a Sophie Germain prime because:$$2 \times 3449+1=6899 \text{ is prime}$$3449 also features in so-called "Golden Semiprimes" and this qualifies it for membership in OEIS A108544:

A108544: primes that are factors of distinct golden semiprimes (A108540).

I posted about these types of semiprimes in Semiprime Factor Ratios way back on the 26th August 2016. In that post I said that:

A golden semiprime is a number that factors to:

• \(p \times q\) where \(p\) and \(q\) are prime
• \( | \,p \, \phi - q \,| <1\) where \(\phi=\dfrac{\sqrt{5}+1}{2} \)

In the case of 3449, it is the \(q\) and \(p=2131\) and the golden semiprime is:$$7349819=2131 \times 3449$$The OEIS mentions 314 sequences in which 3449 makes an appearance and I've only dealt with a few of them here. However, I see 3449 as an important number in my life and didn't want its current occurrence to pass unnoticed.

New Telephone Number

It's always exciting to get a new telephone number because of the properties of the number that may turn up. Having arrived in Australia for a temporary stay, I needed a telephone number and the number that I was given was:$$0451591949 \rightarrow 451591949$$Now this number is prime but has additional "primeness" embedded in it because:$$ \text{Sum of digits is }47 \text{ and prime}\\47 \rightarrow 4 + 7 = 11 \\ 11 \rightarrow 1+1=2 \\ \text{ 2 (the digital root) is prime}$$Now 1949 and 1951 form a pair of twin primes but it turns out that:$$ 451591949 \text{ and } 4511591951 \\ \text{ also form a pair of twin primes}$$Furthermore:$$451591949 \text{ is a Sophie Germain prime} \\ 451591949 \times 2 + 1 = 903183899 \text{ a prime}$$\(451591949\) is also a Chen prime defined as a prime number  \(p\)  such that  \(p+2\)  is either a prime or a semiprime. Jing Run Chen, after which they are named, proved in 1966 that there are infinitely many such primes. Binbin Zhou has proved in 2009 that the Chen primes contain arbitrarily long arithmetic progressions.

So overall I'm very happy with my new prime telephone number even though it will lapse and be discarded once I leave the country. For now though it's mine and prime!

What's Special About 26862?

As my diurnal age today is 26862, I thought it worthy of some detailed analysis. Recently I've started to give such palindromic days posts of their own. For example:

• Palindromic Cyclops Numbers (this is about 26062 in particular)
• 26262: A Special Palindrome
• 26362: Another Special Palindrome
• 26562: A Mid-Millennial Palindrome
• 26662 (this corresponded to my 73rd birthday)

I've also made several posts about palindromes in general:

• Numbers As Sums Of Palindromes
• Remembering Reverse and Add, Palindromes and Trajectories
• L-th Order Palindromes
• Lycrel Numbers
• 22, Reverse and Add

So let's get started on 26862. To begin with it's what I call a five digit “balanced” number. By this I mean a number such that the sum of the first two digits equals the middle digit and the sum of the last two digits equals the middle digit. There are 330 numbers with this property but only 45 of them are palindromes (permalink). The palindromes are:

10101, 11211, 12321, 13431, 14541, 15651, 16761, 17871, 18981, 20202, 21312, 22422, 23532, 24642, 25752, 26862, 27972, 30303, 31413, 32523, 33633, 34743, 35853, 36963, 40404, 41514, 42624, 43734, 44844, 45954, 50505, 51615, 52725, 53835, 54945, 60606, 61716, 62826, 63936, 70707, 71817, 72927, 80808, 81918, 90909

Thus in the case of 26862 we have:$$ \underbrace{2 \, 6}_{2+6=8} \, 8 \, \underbrace{6 \, 2}_{6+2=8}$$The palindromes in particular that have 8 as the central digit are:

• 17871
• 26862
• 35853
• 44844
• 53853
• 62862
• 71871
• 80808

All eight palindromes and thus linked to the famous 888.

About 90% of numbers can be expressed as sum of two palindromes and 26862 is such a number. It can be represented as a sum of two distinct palindromes in 31 different ways (permalink). If the two palindromes don't need to be unique then we can add 13431+13431 for a total of 32. The palidromes are:

[10001+16861], [10101+16761], [10201+16661], [10301+16561], [10401+16461], [10501+16361], [10601+16261], [10701+16161], [10801+16061], [11011+15851], [11111+15751], [11211+15651], [11311+15551], [11411+15451], [11511+15351], [11611+15251], [11711+15151], [11811+15051], [12021+14841], [12121+14741], [12221+14641], [12321+14541], [12421+14441], [12521+14341], [12621+14241], [12721+14141], [12821+14041], [13031+13831], [13131+13731], [13231+13631], [13331+13531], [13431+13431]

However, of these 31, there are only four pairs in which both numbers are prime (permalink). These are:

[10301+16561], [10501+16361], [11311+15551], [11411+15451]

This property of the number qualifies it for membership in OEIS A356854:

A356854

Palindromes that can be written in more than one way as the sum of two distinct palindromic primes.

Here are is the list of sequence members up to 40,000:

282, 484, 858, 888, 21912, 22722, 23832, 24642, 25752, 26662, 26762, 26862, 26962, 27672, 27772, 27872, 27972, 28482, 28782, 28882, 28982, 29692, 29792, 29892, 29992

All numbers can be represented as a sum of three palindromes and there are 190 ways to do so with 26862. I won't list them all here but one example is 161 + 949 + 25752.

The number 26862 is not only symmetric internally but also externally in a number of ways. To begin with it is sandwiched between two primes and is thus the average of the two:$$\underbrace{26861}_{\text{prime}} \, 26862 \, \underbrace{26863}_{\text{prime}}$$Furthermore, it is also a practical number that is the average of the previous practical number (two below it) and the next practical number (two above it). Practical numbers are always even. Thus we have:$$\underbrace{26860}_{\text{practical}} \, \underbrace{26862}_{\text{practical}} \, \underbrace{26864}_{\text{practical}} $$These prime number and practical number properties qualify 26862 for membership in OEIS A209236:

A209236

List of integers m>0 with m-1 and m+1 both prime, and m-2, m, m+2 all practical.

Such numbers are few and far between. Here is the list of sequence members up to 100,000:

4, 6, 18, 30, 198, 462, 1482, 2550, 3330, 4422, 9042, 11778, 26862, 38610, 47058, 60258, 62130, 65538, 69498, 79902, 96222

Even triples of practical numbers are infrequent as can be seen from the initial sequence members of OEIS A287682:

A287682

Triples of practical numbers: numbers n such that n-2, n, n+2 are all practical numbers.

Here are the members up to 40,000:

4, 6, 18, 30, 198, 306, 462, 702, 1482, 2550, 3330, 4422, 5778, 6102, 6498, 9042, 11178, 11778, 14418, 15498, 17298, 17442, 19458, 20862, 21582, 22878, 23322, 23550, 25230, 26622, 26862, 26910, 27378, 30210, 34542, 36738, 38610, 39006, 39102

So these are just a few ways in which 26862 is special and their combination of course makes the number unique.

Heptagonal Numbers

Today I turned 25586 days old and, as it turns out, this is a centered heptagonal number given by the formula:$$
\frac{7 n \; (n-1)}{2}+1 \text{  or  } \frac{7n^2-7n+2}{2}
$$

Figure 1: the initial centred heptagonal numbers

The initial centered heptagonal numbers are given by OEIS A069099 (note that these numbers alternate parity in the pattern odd-even-even-odd):

1, 8, 22, 43, 71, 106, 148, 197, 253, 316, 38 , 463, 547, 638, 736, 841, 953, 1072, 1198, 1331, 1471, 1618, 1772, 1933, 2101, 2276, 2458, 2647, 2843, 3046, 3256, 3473, 3697, 3928, 4166, 4411, 4663, 4922, 5188, 5461, 5741, 6028, 6322, 6623, 6931, 7246, 7568, 7897, 8233, 8576, 8926, 9283, 9647, 10018, 10396, 10781, 11173, 11572, 11978, 12391, 12811, 13238, 13672, 14113, 14561, 15016, 15478, 15947, 16423, 16906, 17396, 17893, 18397, 18908, 19426, 19951, 20483, 21022, 21568, 22121, 22681, 23248, 23822, 24403, 24991, 25586

Numbers Aplenty provides a couple of interesting formulae but nothing concerning how they were derived. The formulae are:$$\sum_{n=1}^\infty \frac{1}{H_n}=\frac{2 \pi \tanh \big ( \frac{\pi}{2\sqrt 7} \big )}{\sqrt 7} \text{  and  } \sum_{n=1}^\infty \frac{H_n}{2^n}=15$$Both are fine-looking formulae where \( \tanh \) is the hyperbolic tangent function defined as:$$ \tanh \theta = \frac{\sinh \theta}{\cosh \theta}=\frac{e^{\theta}-e^{-\theta}}{e^{\theta}+e^{- \theta}}=\frac{e^{2 \theta}-1}{e^{2 \theta}+1}$$A centered heptagonal prime is a centered heptagonal number that is prime. These primes form OEIS A144974 and the initial members of this sequence are:

43, 71, 197, 463, 547, 953, 1471, 1933, 2647, 2843, 3697, 4663, 5741, 8233, 9283, 10781, 11173, 12391, 14561, 18397, 20483, 29303, 29947, 34651, 37493, 41203, 46691, 50821, 54251, 56897, 57793, 65213, 68111, 72073, 76147, 84631, 89041

OEIS A144975 also lists a sequence of centered heptagonal twin prime numbers. Each member of this sequence is associated with its twin e.g. 43 is associated with 41 and 71 is associated with 73 etc. and the initial members of this sequence are:

43, 71, 197, 463, 1933, 5741, 8233, 9283, 11173, 14561, 34651, 41203, 57793, 68111, 84631, 104147, 139301, 168631, 207523, 244861, 307693, 333103, 357281, 415381, 465011, 475273, 506731, 592663, 595547, 607153, 729373, 742211, 781397, 876751

So far I've only focused on centered heptagonal numbers but another category of figurate numbers is the heptagonal numbers defined by the formula:$$\frac{5n^2-3n}{2}$$These numbers comprise OEIS A000566 and the initial members are:

0, 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, 616, 697, 783, 874, 970, 1071, 1177, 1288, 1404, 1525, 1651, 1782, 1918, 2059, 2205, 2356, 2512, 2673, 2839, 3010, 3186, 3367, 3553, 3744, 3940, 4141, 4347, 4558, 4774, 4995, 5221, 5452, 5688 

The parity of heptagonal numbers follows the pattern odd-odd-even-even. Like square numbers, the digital root in base 10 of a heptagonal number can only be 1, 4, 7 or 9. Five times a heptagonal number, plus 1 equals a triangular number.

There is an impressive formula for the sum of the reciprocals of the heptagonal numbers:$$ \begin{align} &\sum_{n=1}^\infty \frac{2}{n(5n-3)} = \frac{1}{15}{\pi}{\sqrt{25-10\sqrt{5}}} \\ +  &\frac{2}{3}\ln(5)+\frac{{1}+\sqrt{5}}{3}\ln\left(\frac{1}{2}\sqrt{10-2\sqrt{5}}\right)+\frac{{1}-\sqrt{5}}{3}\ln\left(\frac{1}{2}\sqrt{10+2\sqrt{5}}\right) \end{align} $$This formula and other related formulae are derived in a June 2008 paper titled Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers. The paper is not for the faint-hearted. Another paper that I stumbled upon when investigating heptagonal numbers was titled Heptagonal Numbers in the Lucas Sequence and Diophantine Equations \(x^2(5x -  3)^2 = 20y^2  \pm 16\). This paper doesn't look as formidable.

However, an article titled Project Euler 61: Find the sum of the only set of six 4-digit figurate numbers with a cyclic property on this site proved to be of the most interest. The article begins:

For some reason Problem 61 of Project Euler is a problem that not so many people have solved compared to the problems in the sixties range.  However, I think that it was a quite approachable problem which was fun to solve.  The problem reads: 

Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygonal) numbers and are generated by the following formulae: 

Triangle: \(P_{3,n}=n(n+1)/2 \) with initial members 1, 3, 6, 10, 15, …
Square: \(P_{4,n}=n^2\) with initial members 1, 4, 9, 16, 25, …
Pentagonal: \(P_{5,n}=n(3n-1)/2 \) with initial members 1, 5, 12, 22, 35, …
Hexagonal: \(P_{6,n}=n(2n-1) \) with initial members 1, 6, 15, 28, 45, …
Heptagonal: \(P_{7,n}=n(5n-3)/2 \) with initial members 1, 7, 18, 34, 55, …
Octagonal: \(P_{8,n}=n(3n-2)  \) with initial members 1, 8, 21, 40, 65, … 

The ordered set of three 4-digit numbers: 8128, 2882, 8281, has three interesting properties: 

1. The set is cyclic, in that the last two digits of each number is the first two digits of the next number (including the last number with the first). 

2. Each polygonal type: triangle (\(P_{3,127}=8128 \)), square (\(P_{4,91}=8281)\), and pentagonal (\(P_{5,44}=2882 \)), is represented by a different number in the set. 

3. This is the only set of 4-digit numbers with this property. 

Find the sum of the only ordered set of six cyclic 4-digit numbers for which each polygonal type: triangle, square, pentagonal, hexagonal, heptagonal, and octagonal, is represented by a different number in the set.

The author of the article provides the code he used to solve this problem. As he said, he used a brute force approach as I probably would if I were attempting it in SageMath (which I haven't tried as yet). Anyway, to cut to the chase, the six cyclic 4-digit numbers are:

1281, 8128, 2882, 8256, 5625, 2512

2512 is the heptagonal number, 1281 is octagonal, 8128 is hexagonal, 2882 is pentagonal, 8256 is triangular and 5625 is a square number (and also a centered octagonal number). 

This article got me curious about Project Euler so I decided to investigate further. Here is what I found:

About Project Euler
Leonhard Euler (1707-1783) 

What is Project Euler? 

Project Euler is a series of challenging mathematical/computer programming problems that will require more than just mathematical insights to solve. Although mathematics will help you arrive at elegant and efficient methods, the use of a computer and programming skills will be required to solve most problems. 

The motivation for starting Project Euler, and its continuation, is to provide a platform for the inquiring mind to delve into unfamiliar areas and learn new concepts in a fun and recreational context.

Who are the problems aimed at? 

The intended audience include students for whom the basic curriculum is not feeding their hunger to learn, adults whose background was not primarily mathematics but had an interest in things mathematical, and professionals who want to keep their problem solving and mathematics on the cutting edge.

Can anyone solve the problems? 

The problems range in difficulty and for many the experience is inductive chain learning. That is, by solving one problem it will expose you to a new concept that allows you to undertake a previously inaccessible problem. So the determined participant will slowly but surely work his/her way through every problem.

What next? 

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However, as the problems are challenging then you may wish to view the Problems before registering.

There are currently 656 problems listed and I thought it would be interesting to try to solve some of them using SageMath, so I signed up to the site. The first problem I tried was a very simple one but hey, it's a start. Here was the problem:

Each new term in the Fibonacci sequence is generated by adding the previous two terms. By starting with 1 and 2, the first 10 terms will be: 

1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ... 

By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms.

The answer turns out to be 4613732 and here is the SageMath code that I used to arrive at the solution:

n=0
sum=0
while fibonacci(n)<4000000:
    if fibonacci(n) % 2==0:
        sum+=fibonacci(n)
    n=n+1
print sum

Figure 2: feedback from Project Euler

You might think it's cheating to use the fibonacci function in SageMath but it's easy enough to generate the Fibonacci sequence without resorting to it. However, it's available so I used it. Figure 2 shows the feedback I received from Project Euler. I'm encouraged to try to solve some of the other 655 problems remaining, hopefully with a higher difficulty rating.