Palindromic prime

Key takeaways

Palindromic prime belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Palindromic prime to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Palindromic prime from memory before moving on to harder problems.

Reference excerpt

In mathematics, a palindromic prime (sometimes called a palprime) is a prime number that is also a palindromic number. Palindromicity depends on the base of the number system and its notational conventions, while primality is independent of such concerns. The first few decimal palindromic primes are:

2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, … (sequence A002385 in the OEIS) Except for 11, all palindromic primes have an odd number of digits, because the divisibility test for 11 tells us that every palindromic number with an even number of digits is a multiple of 11. It is not known if there are infinitely many palindromic primes in base 10. For any base, almost all palindromic numbers are composite, i.e. the ratio between palindromic composites and all palindromes less than n tends to 1. A few decorative examples do however exist; in base 10 the following are primes: 11, 122333221, and 1223334444555554444333221. So are: 13331, and 12233355555333221. For a large example, consider:

101888529 − 10944264 − 1, which has 1,888,529 digits. It was found on 18 October 2021 by Ryan Propper and Serge Batalov. All decimal repunits are palindromic numbers, some of which are prime. Among them, the largest currently known candidate for primality is the probable prime 10 8177207 − 1 9 {\displaystyle {\tfrac {10^{8177207}-1}{9}}} , which has 8,177,207 digits.

Other bases In binary, the palindromic primes include the Mersenne primes and the Fermat primes. All binary palindromic primes except binary 11 (decimal 3) have an odd number of digits; those palindromes with an even number of digits are divisible by 3. The sequence of binary palindromic primes begins (in binary):

11, 101, 111, 10001, 11111, 1001001, 1101011, 1111111, 100000001, 100111001, 110111011, ... (sequence A117697 in the OEIS) Any number that can be expressed as a repetition of just one digit d in some base must trivially be palindromic in that base and must be a multiple of d in every base. Accordingly, no number that consists only of a string of repetitions of the same digit in at least one base, can be a prime unless it is a string of 1s in that base. Furthermore, the string must be of prime length, otherwise, if the string is of length mXn, it is divisible by strings of lengths m and n in that base. For example 111111111111111 (15 digits) is divisible by 111 and 11111 in that base. If a number m can be expressed as a string of prime length to some base, such a number may or may not be prime, but commonly is not; for example, to base 10, there are only three such numbers of length less than 100 (1 is by definition, not prime). The three are: 11 (length 2), 1111111111111111111 (length 19), and 11111111111111111111111 (length 23)

Property Due to the superstitious significance of the numbers it contains, the palindromic prime 1000000000000066600000000000001 is known as Belphegor's Prime, named after Belphegor, one of the seven princes of Hell. Belphegor's Prime consists of the number 666, on either side enclosed by thirteen zeroes and a one. Belphegor's Prime is an example of a beastly palindromic prime in which a prime p is palindromic with 666 in the center. Another beastly palindromic prime is 700666007. Ribenboim defines a triply palindromic prime as a prime p for which: p is a palindromic prime with q digits, where q is a palindromic prime with r digits, where r is also a palindromic prime. For example, p = 1011310 + 4661664×105652 + 1, which has q = 11311 digits, and 11311 has r = 5 digits. The first (base-10) triply palindromic prime is the 11-digit number 10000500001. It is possible for a triply palindromic prime in base 10 to also be palindromic in another base, such as base 2, but it would be incredibly remarkable if it were also a triply palindromic prime in that base as well.

Bounds Let P ( x ) {\displaystyle {\mathcal {P}}(x)} be the set of all palindromic numbers less than x. Banks, Hart, and Sakata (2004) have shown that, using big-O notation, the number of palindromic primes less than x is O ( # P ( x ) log ⁡ log ⁡ x log ⁡ x ) {\displaystyle O\left(\#{\mathcal {P}}(x){\frac {\log \log x}{\log x}}\right)} . Col (2006) further improved this bound to O ( # P ( x ) log ⁡ x ) {\displaystyle O\left({\frac {\#{\mathcal {P}}(x)}{\log x}}\right)} .

Worked examples

Example 1 — a first encounter with Palindromic prime

Start with the simplest possible case. Write down what Palindromic prime claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Palindromic prime before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Palindromic prime ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Palindromic prime

In research: Palindromic prime appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Palindromic prime in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Palindromic prime is common in secondary-school and first-year university syllabi. It links to neighbouring topics Base-dependent integer sequences, Classes of prime numbers, Palindromes, so understanding it makes those chapters shorter.
In everyday life: Look for Palindromic prime outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Palindromic prime in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Palindromic prime means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Palindromic prime out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Palindromic prime in simple terms?

In mathematics, a palindromic prime (sometimes called a palprime) is a prime number that is also a palindromic number. Palindromicity depends on the base of the number system and its notational conventions, while primality is independent of such concerns.

Why does Palindromic prime matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Palindromic prime?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Palindromic prime.