mathematics

Pillai sequence

Pillai sequence 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 Pillai sequence rather than just read about it. In short: The Pillai sequence is the sequence of integers that have record numbers of terms in their greedy representations as sums of prime numbers (and one). It is named after Subbayya Sivasankaranarayana Pillai, who first defined it in 1930.

Key takeaways

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

Reference excerpt

The Pillai sequence is the sequence of integers that have record numbers of terms in their greedy representations as sums of prime numbers (and one). It is named after Subbayya Sivasankaranarayana Pillai, who first defined it in 1930. It would follow from Goldbach's conjecture that every integer greater than one can be represented as a sum of at most three prime numbers. However, finding such a representation could involve solving instances of the subset sum problem, which is computationally difficult. Instead, Pillai considered the following simpler greedy algorithm for finding a representation of n {\displaystyle n} as a sum of primes: choose the first prime in the sum to be the largest prime p {\displaystyle p} that is at most n {\displaystyle n} , and then recursively construct the remaining sum recursively for n − p {\displaystyle n-p} . If this process reaches zero, then it halts; if it reaches one instead of zero, then it must include one in the sum (even though it is not prime), and then halt. For instance, this algorithm represents 122 as 113 + 7 + 2, even though the shorter representations 61 + 61 or 109 + 13 are also possible. The n {\displaystyle n} th number in the Pillai sequence is the smallest number whose greedy representation as a sum of primes (and one) requires n {\displaystyle n} terms. These numbers are

0, 1, 4, 27, 1354, 401429925999155061, ... (sequence A066352 in the OEIS). Each number a ( n ) {\displaystyle a(n)} in the sequence is the sum of the previous number a ( n − 1 ) {\displaystyle a(n-1)} with a prime number p {\displaystyle p} , the smallest prime whose following prime gap is larger than a ( n − 1 ) {\displaystyle a(n-1)} . For instance, the number 27 in the sequence is 4 + 23, and the first prime gap larger than 4 is the one between 23 and 29. Because the prime numbers become less dense as they become larger (as quantified by the prime number theorem), there is always a prime gap larger than any term in the Pillai sequence, so the sequence continues to an infinite number of terms. However, the terms in the sequence grow very rapidly; Cramér's conjecture implies the sequence grows tetrationally. It has been estimated that expressing the next term in the sequence would require "hundreds of millions of digits".

References

Worked examples

Example 1 — a first encounter with Pillai sequence

Start with the simplest possible case. Write down what Pillai sequence 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 Pillai sequence 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 Pillai sequence 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 Pillai sequence

In research: Pillai sequence 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 Pillai sequence 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: Pillai sequence is common in secondary-school and first-year university syllabi. It links to neighbouring topics Integer sequences, Prime numbers, so understanding it makes those chapters shorter.
In everyday life: Look for Pillai sequence 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 Pillai sequence in 20 minutes

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

Frequently asked questions

What is Pillai sequence in simple terms?

Why does Pillai sequence 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 Pillai sequence?

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 Pillai sequence.