Proth Number Checker

Check whether a decimal integer is k·2^m+1 with m≥1 and positive odd k<2^m, without requiring primality; input is capped at 200 characters.

Description

A Proth number has the form k·2^m+1 with positive odd k smaller than 2^m.

A Proth number has the form k·2^m+1 with positive odd k smaller than 2^m. Being a Proth number and being a Proth prime are separate properties.^[1]

The tool returns isProth. The indexing and output conventions below determine how to read these values.

When to use Proth Number Checker

Use this for the following task: Check whether a decimal integer is k·2^m+1 with m≥1 and positive odd k<2^m, without requiring primality; input is capped at 200 characters. Use the input and output conventions below to reproduce the intended mathematical quantity.

How to use Proth Number Checker and understand its results

Choose the required indexes, integers or sequence entries. Preserve decimal-string encodings where the field table specifies strings.
Enter n according to the field table, then run the tool.
Read isProth. A true flag means the candidate meets the stated property; false means it does not.
If the flag is false, review the candidate and definition. A well-formed nonmember is different from a rejected input.

Input field	Meaning	Value type and bounds	Requirement
n	Nonnegative decimal integer	string (integer/arbitrary); maximum text length 200; pattern ^(0\|[1-9][0-9]*)$	Required

Read the decimal input with BigInt. Exclude n<3 and even n, then remove factors of two from n−1 and compare the remaining odd factor with the extracted power of two.

Proth ⇔ n−1=k×2^m, m≥1, odd k>0, k<2^m

The symbols and boundary conditions refer to the supplied fields and the implementation described above.

Output field	Meaning	Value type
isProth	Whether n is a Proth number	boolean

json

{
  "input": {
    "n": "13"
  },
  "output": {
    "isProth": true
  }
}

Limitations and assumptions

The tool checks the algebraic form only. It does not apply Proth’s primality theorem, and input text is capped at 200 characters.

Use the declared integer bounds and string formats. Large exact outputs remain decimal strings; numerical approximations use ordinary floating-point arithmetic. A rejected input is different from a valid zero count or false membership flag.

Alternative or Complementary methods

Choose Woodall Number Checker when you need its operation: Check whether a nonnegative decimal integer has the form k·2^k−1 for k≥1, without requiring primality; input is capped at 200 characters. Choose Fermat Number Checker for the complementary operation: Check whether a nonnegative decimal integer has the form 2^(2^k)+1 for k≥0, including composite Fermat numbers; input is capped at 200 characters.

Background reference: OEIS A080075.

References

OEIS A080075 — OEIS Foundation

Similar or alternative tools

Woodall Number Checker
Check whether a nonnegative decimal integer has the form k·2^k−1 for k≥1, without requiring primality; input is capped at 200 characters.
Fermat Number Checker
Check whether a nonnegative decimal integer has the form 2^(2^k)+1 for k≥0, including composite Fermat numbers; input is capped at 200 characters.