mathematics

Good prime

Good 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 Good prime rather than just read about it. In short: A good prime is a prime number whose square is greater than the product of any two primes at the same number of positions before and after it in the sequence of primes. That is, good prime satisfies the inequality p n 2 > p n − i ⋅ p n + i {\displaystyle p_{n}^{2}>p_{n-i}\cdot p_{n+i}} for all 1 ≤ i ≤ n−1, where pk is the kth prime.

Key takeaways

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

Reference excerpt

p n 2 > p n − i ⋅ p n + i {\displaystyle p_{n}^{2}>p_{n-i}\cdot p_{n+i}}

for all 1 ≤ i ≤ n−1, where pk is the kth prime. Example: the first primes are 2, 3, 5, 7 and 11. Since for 5 both the conditions

5 2 > 3 ⋅ 7 {\displaystyle 5^{2}>3\cdot 7}

5 2 > 2 ⋅ 11 {\displaystyle 5^{2}>2\cdot 11}

are fulfilled, 5 is a good prime. There are infinitely many good primes. The first good primes are:

5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 97, 101, 127, 149, 179, 191, 223, 227, 251, 257, 269, 307, 311, 331, 347, 419, 431, 541, 557, 563, 569, 587, 593, 599, 641, 727, 733, 739, 809, 821, 853, 929, 937, 967 (sequence A028388 in the OEIS). An alternative version takes only i = 1 in the definition. With that there are more good primes:

5, 11, 17, 29, 37, 41, 53, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 157, 163, 173, 179, 191, 197, 211, 223, 227, 239, 251, 257, 263, 269, 277, 281, 307, 311, 331, 347, 367, 373, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499, 521, 541, 557, 563, 569, 587, 593, 599, 607, 613, 617, 631, 641, 653, 659, 673, 701, 719, 727, 733, 739, 751, 757, 769, 787, 809, 821, 827, 853, 857, 877, 881, 907, 929, 937, 947, 967, 977, 991 (sequence A046869 in the OEIS).

References

Worked examples

Example 1 — a first encounter with Good prime

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

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

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

Frequently asked questions

What is Good prime in simple terms?

A good prime is a prime number whose square is greater than the product of any two primes at the same number of positions before and after it in the sequence of primes. That is, good prime satisfies the inequality p n 2 > p n − i ⋅ p n + i {\displaystyle p_{n}^{2}>p_{n-i}\cdot p_{n+i}} for all 1 ≤…

Why does Good 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 Good 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 Good prime.