mathematics

Newman–Shanks–Williams prime

Newman–Shanks–Williams 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 Newman–Shanks–Williams prime rather than just read about it. In short: In mathematics, a Newman–Shanks–Williams prime (NSW prime) is a prime number which can be written in the form S 2 m + 1 = ( 1 + 2 ) 2 m + 1 + ( 1 − 2 ) 2 m + 1 2 {\displaystyle S_{2m+1}={\frac {\left(1+{\sqrt {2}}\right)^{2m+1}+\left(1-{\sqrt {2}}\right)^{2m+1}}{2}}} . NSW primes were first described by Morris Newman, Daniel Shanks and Hugh C.

Key takeaways

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

Reference excerpt

In mathematics, a Newman–Shanks–Williams prime (NSW prime) is a prime number which can be written in the form

S 2 m + 1 = ( 1 + 2 ) 2 m + 1 + ( 1 − 2 ) 2 m + 1 2 {\displaystyle S_{2m+1}={\frac {\left(1+{\sqrt {2}}\right)^{2m+1}+\left(1-{\sqrt {2}}\right)^{2m+1}}{2}}} . NSW primes were first described by Morris Newman, Daniel Shanks and Hugh C. Williams in 1981 during their study of finite simple groups with square order. The first few NSW primes are 7, 41, 239, 9369319, 63018038201, . . . (sequence A088165 in the OEIS), corresponding to the indices 3, 5, 7, 19, 29, . . . (sequence A005850 in the OEIS). The sequence S {\displaystyle S} alluded to in the formula can be described by the following recurrence relation:

S 0 = 1 {\displaystyle S_{0}=1\,}

S 1 = 1 {\displaystyle S_{1}=1\,}

S n = 2 S n − 1 + S n − 2 for all n ≥ 2. {\displaystyle S_{n}=2S_{n-1}+S_{n-2}\qquad {\text{for all }}n\geq 2.}

The first few terms of the sequence are 1, 1, 3, 7, 17, 41, 99, . . . (sequence A001333 in the OEIS). Each term in this sequence is half the corresponding term in the sequence of companion Pell numbers. These numbers also appear in the continued fraction convergents to √2.

References

External links The Prime Glossary: NSW number

Worked examples

Example 1 — a first encounter with Newman–Shanks–Williams prime

Start with the simplest possible case. Write down what Newman–Shanks–Williams 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 Newman–Shanks–Williams 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 Newman–Shanks–Williams 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 Newman–Shanks–Williams prime

In research: Newman–Shanks–Williams 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 Newman–Shanks–Williams 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: Newman–Shanks–Williams prime is common in secondary-school and first-year university syllabi. It links to neighbouring topics Classes of prime numbers, Unsolved problems in number theory, so understanding it makes those chapters shorter.
In everyday life: Look for Newman–Shanks–Williams 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 Newman–Shanks–Williams prime in 20 minutes

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

Frequently asked questions

What is Newman–Shanks–Williams prime in simple terms?

In mathematics, a Newman–Shanks–Williams prime (NSW prime) is a prime number which can be written in the form S 2 m + 1 = ( 1 + 2 ) 2 m + 1 + ( 1 − 2 ) 2 m + 1 2 {\displaystyle S_{2m+1}={\frac {\left(1+{\sqrt {2}}\right)^{2m+1}+\left(1-{\sqrt {2}}\right)^{2m+1}}{2}}} . NSW primes were first describ…

Why does Newman–Shanks–Williams 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 Newman–Shanks–Williams 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 Newman–Shanks–Williams prime.