mathematics

Truncatable prime

Truncatable prime is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Truncatable prime rather than just read about it. In short: In number theory, a left-truncatable prime is a prime number which, in a given base, contains no 0, and if the leading ("left") digit is successively removed, then all resulting numbers are prime. For example, 9137, since 9137, 137, 37 and 7 are all prime.

Key takeaways

Truncatable 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 Truncatable prime to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Truncatable prime from memory before moving on to harder problems.

Reference excerpt

In number theory, a left-truncatable prime is a prime number which, in a given base, contains no 0, and if the leading ("left") digit is successively removed, then all resulting numbers are prime. For example, 9137, since 9137, 137, 37 and 7 are all prime. Decimal representation is often assumed and always used in this article. A right-truncatable prime is a prime which remains prime when the last ("right") digit is successively removed. 7393 is an example of a right-truncatable prime, since 7393, 739, 73, and 7 are all prime. A left-and-right-truncatable prime is a prime which remains prime if the leading ("left") and last ("right") digits are simultaneously successively removed down to a one- or two-digit prime. 1825711 is an example of a left-and-right-truncatable prime, since 1825711, 82571, 257, and 5 are all prime. In base 10, there are exactly 4260 left-truncatable primes, 83 right-truncatable primes, and 920,720,315 left-and-right-truncatable primes.

History Author Leslie E. Card in early volumes of the Journal of Recreational Mathematics (which started its run in 1968) considered a topic close to that of right-truncatable primes, calling sequences that by adding digits to the right in sequence to an initial number not necessarily prime snowball primes. Discussion of the topic dates to at least November 1969 issue of Mathematics Magazine, where truncatable primes were called prime primes by two co-authors (Murray Berg and John E. Walstrom).

Decimal truncatable primes There are 4260 left-truncatable primes:

2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 113, 137, 167, 173, 197, 223, 283, 313, 317, 337, 347, 353, 367, 373, 383, 397, 443, 467, 523, 547, 613, 617, 643, 647, 653, 673, 683, 743, 773, 797, 823, 853, 883, 937, 947, 953, 967, 983, 997, ... (sequence A024785 in the OEIS) The largest is the 24-digit 357686312646216567629137. There are 83 right-truncatable primes. The complete list:

2, 3, 5, 7, 23, 29, 31, 37, 53, 59, 71, 73, 79, 233, 239, 293, 311, 313, 317, 373, 379, 593, 599, 719, 733, 739, 797, 2333, 2339, 2393, 2399, 2939, 3119, 3137, 3733, 3739, 3793, 3797, 5939, 7193, 7331, 7333, 7393, 23333, 23339, 23399, 23993, 29399, 31193, 31379, 37337, 37339, 37397, 59393, 59399, 71933, 73331, 73939, 233993, 239933, 293999, 373379, 373393, 593933, 593993, 719333, 739391, 739393, 739397, 739399, 2339933, 2399333, 2939999, 3733799, 5939333, 7393913, 7393931, 7393933, 23399339, 29399999, 37337999, 59393339, 73939133 (sequence A024770 in the OEIS) The largest is the 8-digit 73939133. All primes above 5 end with digit 1, 3, 7 or 9, so a right-truncatable prime can only contain those digits after the leading digit. There are 920,720,315 left-and-right-truncatable primes:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 127, 131, 137, 139, 151, 157, 173, 179, 223, 227, 229, 233, 239, 251, 257, 271, 277, 331, 337, 353, 359, 373, 379, 421, 431, 433, 439, 457, 479, 521, 523, 557, 571, 577, 631, 653, 659, 673, 677, 727, 733, 739, 751, 757, 773, 821, 823, 827, 829, 839, 853, 857, 859, 877, 929, 937, 953, 971, 977, 1117, 1171, 1193, 1231, 1237, 1291, 1297, 1319, 1373, 1433, 1439, 1471, 1531, 1597, 1613, 1619, ... (sequence A077390 in the OEIS) There are 331,780,864 left-and-right-truncatable primes with an odd number of digits. The largest is the 97-digit prime 7228828176786792552781668926755667258635743361825711373791931117197999133917737137399993737111177. There are 588,939,451 left-and-right-truncatable primes with an even number of digits. The largest is the 104-digit prime 91617596742869619884432721391145374777686825634291523771171391111313737919133977331737137933773713713973. There are 15 primes which are both left-truncatable and right-truncatable. They have been called two-sided primes. The complete list:

2, 3, 5, 7, 23, 37, 53, 73, 313, 317, 373, 797, 3137, 3797, 739397 (sequence A020994 in the OEIS) A left-truncatable prime is called restricted if all of its left extensions are composite i.e. there is no other left-truncatable prime of which this prime is the left-truncated "tail". Thus 7937 is a restricted left-truncatable prime because the nine 5-digit numbers ending in 7937 are all composite, whereas 3797 is a left-truncatable prime that is not restricted because 33797 is also prime. There are 1442 restricted left-truncatable primes:

2, 5, 773, 3373, 3947, 4643, 5113, 6397, 6967, 7937, 15647, 16823, 24373, 33547, 34337, 37643, 56983, 57853, 59743, 62383, 63347, 63617, 69337, 72467, 72617, 75653, 76367, 87643, 92683, 97883, 98317, ... (sequence A240768 in the OEIS) Similarly, a right-truncatable prime is called restricted if all of its right extensions are composite. There are 27 restricted right-truncatable primes:

53, 317, 599, 797, 2393, 3793, 3797, 7331, 23333, 23339, 31193, 31379, 37397, 73331, 373393, 593993, 719333, 739397, 739399, 2399333, 7393931, 7393933, 23399339, 29399999, 37337999, 59393339, 73939133 (sequence A239747 in the OEIS)

Other bases While the primality of a number does not depend on the numeral system used, truncatable primes are defined only in relation with a given base. A variation involves removing 2 or more decimal digits at a time. This is mathematically equivalent to using base 100 or a larger power of 10, with the restriction that base 10n digits must be at least 10n−1, in order to match a decimal n-digit number with no leading 0.

Worked examples

Example 1 — a first encounter with Truncatable prime

Start with the simplest possible case. Write down what Truncatable 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 Truncatable 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 Truncatable 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 Truncatable prime

In research: Truncatable 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 Truncatable 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: Truncatable prime is common in secondary-school and first-year university syllabi. It links to neighbouring topics Base-dependent integer sequences, Classes of prime numbers, so understanding it makes those chapters shorter.
In everyday life: Look for Truncatable 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.

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How to study Truncatable prime in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Truncatable 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 Truncatable prime out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Truncatable prime in simple terms?

Why does Truncatable 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 Truncatable 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 Truncatable prime.