Type-Level Prime Number Checker in TypeScript

This challenge asks you to implement a type-level function in TypeScript that determines whether a given number, represented as a type, is a prime number. Type-level programming allows us to perform computations and logic directly within the type system, enabling powerful compile-time validations and optimizations. This exercise will test your understanding of conditional types, recursive types, and potentially distributive conditional types in TypeScript.

Problem Description

You need to create a type IsPrime<N> where N is a number represented as a type (e.g., 1 is represented as 1 extends 1 ? true : false). The IsPrime<N> type should resolve to true if N is a prime number and false otherwise.

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Therefore, to determine if a number N is prime, you need to check if it's divisible by any number from 2 up to the square root of N. Since we're working at the type level, we'll represent divisibility using conditional types and recursion.

Key Requirements:

The solution must be purely type-level; no runtime code is allowed.
The solution should handle numbers from 2 upwards. Numbers less than 2 should be considered non-prime.
The solution should be efficient in terms of type complexity. Avoid unnecessary type recursion.
The solution should correctly identify prime numbers.

Expected Behavior:

IsPrime<2> should resolve to true.
IsPrime<3> should resolve to true.
IsPrime<4> should resolve to false.
IsPrime<5> should resolve to true.
IsPrime<6> should resolve to false.
IsPrime<7> should resolve to true.
IsPrime<8> should resolve to false.
IsPrime<9> should resolve to false.
IsPrime<10> should resolve to false.
IsPrime<11> should resolve to true.
IsPrime<1> should resolve to false.
IsPrime<0> should resolve to false.

Examples

Example 1:

Input: IsPrime<5>
Output: true
Explanation: 5 is only divisible by 1 and 5, therefore it is prime.

Example 2:

Input: IsPrime<4>
Output: false
Explanation: 4 is divisible by 2, therefore it is not prime.

Example 3:

Input: IsPrime<1>
Output: false
Explanation: 1 is not a prime number.

Constraints

N will be a number represented as a type, typically a union of types representing the number.
The solution should be reasonably performant in terms of type checking time. Extremely complex or deeply recursive type definitions can lead to compiler errors or very long compilation times.
The solution should handle numbers up to a reasonable limit (e.g., 20). While there's no strict upper bound, extremely large numbers will likely cause type checking issues.

Notes

Consider using distributive conditional types to iterate through potential divisors.
You can represent divisibility by checking if N is assignable to DivisibleBy<N, D>, where DivisibleBy<N, D> is a type that checks if N is divisible by D.
Think about how to efficiently calculate the square root of N at the type level (you don't need to calculate the exact square root, just a value close enough to stop the divisibility checks). You can use a type that represents the integer part of the square root.
Start with smaller prime numbers (2, 3, 5, 7) and gradually increase the complexity.
Remember that TypeScript's type system has limitations. Extremely complex type-level computations can lead to compiler errors. Strive for clarity and efficiency.

Type-Level Prime Number Checker in TypeScript

Problem Description

Key Requirements:

The solution must be purely type-level; no runtime code is allowed.

The solution should handle numbers from 2 upwards. Numbers less than 2 should be considered non-prime.

The solution should be efficient in terms of type complexity. Avoid unnecessary type recursion.

The solution should correctly identify prime numbers.

Expected Behavior:

IsPrime<2> should resolve to true.

IsPrime<3> should resolve to true.

IsPrime<4> should resolve to false.

IsPrime<5> should resolve to true.

IsPrime<6> should resolve to false.

IsPrime<7> should resolve to true.

IsPrime<8> should resolve to false.

IsPrime<9> should resolve to false.

IsPrime<10> should resolve to false.

IsPrime<11> should resolve to true.

IsPrime<1> should resolve to false.

IsPrime<0> should resolve to false.

Constraints

N will be a number represented as a type, typically a union of types representing the number.

The solution should be reasonably performant in terms of type checking time. Extremely complex or deeply recursive type definitions can lead to compiler errors or very long compilation times.

The solution should handle numbers up to a reasonable limit (e.g., 20). While there's no strict upper bound, extremely large numbers will likely cause type checking issues.

Notes

Consider using distributive conditional types to iterate through potential divisors.

You can represent divisibility by checking if N is assignable to DivisibleBy<N, D>, where DivisibleBy<N, D> is a type that checks if N is divisible by D.

Think about how to efficiently calculate the square root of N at the type level (you don't need to calculate the exact square root, just a value close enough to stop the divisibility checks). You can use a type that represents the integer part of the square root.

Start with smaller prime numbers (2, 3, 5, 7) and gradually increase the complexity.

Remember that TypeScript's type system has limitations. Extremely complex type-level computations can lead to compiler errors. Strive for clarity and efficiency.