Nearest Prime Number Calculator

Calculator

Integer

Whole numbers only, positive or negative.

Search mode Nearest prime Previous prime only Next prime only

Choose how the prime is selected.

Tie rule Return both primes Prefer lower prime Prefer higher prime

Used when primes are equally distant.

Search limit (radius)

Maximum steps away from the input.

Show checked values preview

May be slower for large limits.

Calculate

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Example data

Formula used

A prime number is an integer greater than 1 with exactly two divisors: 1 and itself. To test primality, the calculator checks divisibility up to √n, skipping obvious composites.

To find the nearest prime to n, the search evaluates candidates n−k and n+k for increasing k, then applies the selected tie rule when distances match.

How to use this calculator

1. Enter any whole number in the Integer field.
2. Select a Search mode: nearest, previous only, or next only.
3. If using nearest, choose a Tie rule for equal distances.
4. Set a Search limit to cap how far the scan can go.
5. Press Calculate to display the result above the form.
6. Use Download CSV or Download PDF for a saved report.

Prime proximity insights

1) What “nearest prime” means

For a given integer n, the nearest prime is the prime with the smallest absolute difference |p−n|. When two primes are equally distant, this calculator can return both or apply a consistent preference toward the lower or higher prime.

2) Why primes appear “less often”

Prime density decreases as numbers grow. A widely used approximation is that the proportion of primes near n is about 1 / ln(n). This helps explain why large inputs may require scanning more candidates to find the next prime.

3) Typical gap sizes you can expect

Around large values, a practical rule of thumb is that the average gap between consecutive primes is roughly ln(n). For example, near one million, ln(1,000,000) ≈ 13.8, so nearest primes are often within a few tens of steps, though gaps can be larger.

4) How the search stays efficient

The calculator tests primality using trial division up to √n and skips obvious composites by checking only candidates of the form 6k ± 1 after handling 2 and 3. This reduces checks dramatically for typical classroom and engineering ranges.

5) Direction and tie-rule controls

“Previous” and “Next” modes are helpful when you need a one-sided bound, such as selecting a prime modulus above a threshold. In “Nearest” mode, the tie rule matters for inputs like 49, where 47 and 53 are equally close.

6) Interpreting the distance value

Distance is the absolute gap between your input and the returned prime(s). A distance of 1 indicates an adjacent prime, while larger distances highlight prime gaps. Reviewing both previous and next primes provides useful context even when only one nearest result is returned.

7) Choosing a safe search limit

The radius limit caps how far the scan can move away from the input. For everyday work, limits between 10,000 and 200,000 are usually ample. For very large inputs, increasing the limit may be needed, but it can also increase compute time.

8) Practical uses and record keeping

Nearest primes support tasks like hashing experiments, modular arithmetic examples, and testing numeric properties around boundaries. The built-in CSV and PDF exports help you document inputs, chosen options, and final primes for audits, homework submissions, or reproducible notes.

FAQs

1) What happens if I enter a negative number?

Primes are defined for integers greater than 1. The calculator still accepts negatives, then searches forward to the first prime at or above 2 when needed.

2) Why can two nearest primes be returned?

If the input sits exactly midway between consecutive primes, both have the same distance. Selecting “Return both primes” shows the pair; other tie rules choose one consistently.

3) How accurate is the primality check?

The test is exact for the numbers this server can represent as integers. It uses deterministic trial division up to the square root, so the result is mathematically correct within that range.

4) What does “search limit” control?

It caps how many steps the scan can move away from your input while looking for primes. If no prime is found within the limit, the calculator reports that outcome.

5) Why does performance vary by input size?

Two effects matter: primes thin out as numbers grow, and primality checks become heavier because √n increases. Larger numbers may require more candidate checks and more divisions.

6) When should I use “previous” or “next” mode?

Use “previous” when you need a prime below a threshold, and “next” when you need a prime above it. This is common in modular arithmetic examples and selecting safe bounds.

7) Do the downloads include my selected options?

Yes. The exported report records the input, mode, tie rule, search limit, the nearest prime(s), distance, neighbors, and notes so you can reproduce the same result later.