On Frustrating Primes

7775353777 is a prime number.  It has ten digits and it looks like, on first inspection, that the second five digits reflect the first.  But a number cannot be prime and have such a property — and on closer inspection we can see that there is, indeed, no such reflection.  Such primes can be considered 'frustrating' because of the absence of this visually eye-catching, aesthetically pleasing pattern they appear, on the initial face of it, to promise.

A prime, therefore, is said to be frustrating if the first half of its digits occur again in the second half, but without a) the order repeating (i.e. abc‖abc, because this is divisible by 1001 [1] and thus not prime), or b) the order reflecting (i.e. abc‖cba, because every palindrome with an even number of digits is divisible by 11 and thus not prime).

As such, a frustrating prime has an even number of digits, and contains at least six digits of which at least two are distinct [2].  Of the 68,906 six-digit primes [3], 265 are frustrating (collated in the image above).  The possible arrangements of six-digits with the first three reoccurring:

aaa‖···

⇒ aaa‖aaa : Not prime ∵  divisible by 11

aab‖···

⇒ aab‖aab : Not prime ∵ divisible by 1001

⇒ aab‖aba : Possibly a frustrating prime (n = 10)

⇒ aab‖baa : Not prime ∵ divisible by 11

aba‖···

⇒ aba‖aab : Possibly a frustrating prime (n = 7)

⇒ aba‖aba : Not prime ∵ divisible by 1001

⇒ aba‖baa : Possibly a frustrating prime (n = 10)

abb‖···

⇒ abb‖abb : Not prime ∵ divisible by 1001

⇒ abb‖bab : Possibly a frustrating prime (n = 9)

⇒ abb‖bba : Not prime ∵ divisible by 11

abc‖···

⇒ abc‖abc : Not prime ∵ divisible by 1001

⇒ abc‖acb : Possibly a frustrating prime (n = 66)

⇒ abc‖bac : Possibly a frustrating prime (n = 58)

⇒ abc‖bca : Possibly a frustrating prime (n = 68)

⇒ abc‖cab : Possibly a frustrating prime (n = 37)

⇒ abc‖cba : Not prime ∵ divisible by 11

The Frustrating Primes are recorded as sequence A297994 in the On-line Encyclopedia of Integer Sequences (OEIS), from which you can find this table of Frustrating Primes a_n for n = 1 to 10,000. 

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The learning capital to be accrued from exploring the idea of frustrating primes, is especially evident in terms of the development of students' mathematical maturity.  The above sketch of the underlying premise of frustrating primes informs a 'thinking through' of how as teachers we could use the idea with (alongside) students, how we could approach the problem with students, structurally and pedagogically, and what questions can/should be asked of our students, should they not yet be able to ask them for themselves.

Considering the idea through this problem will encourage depth and maturation, and give us as teachers the opportunity, as I wrote about in a previous post, to 'model for [students] what it is to be mathematically mature, to behave in a mathematically mature manner when we are befuddled by the beautiful sixes and sevens of uncertainty.'   Students will not only deepen their sense of number and their appreciation of prime numbers, at the same time as being able to apply divisibility tests, consider prime factors, etc., but they may also get a kick out of doing something a little more mathematically novel than they are typically used to.

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Notes & (Select) Links:

[1]  \(\left( {1000 \times \overline {abc} } \right) + \left( {1 \times \overline {abc} } \right)\)

[2]  For a two-digit number to be a frustrating prime the first half of the digits would have to occur in the second, and this would thus be a number of the form a‖a, which is clearly a composite number divisible by 11.  Similarly, for a four-digit number to be a frustrating prime the first half of the digits would have to occur in the second, meaning that the number would thus be of the form ab‖ab or ab‖ba, which are both composite as the former is divisible by 101 (100×ab+1×ab) and the latter is an even palindrome and thus divisible by 11.

[3] For a full list (via Wolfram Alpha)