# Queries for maximum difference between prime numbers in given ranges

• Difficulty Level : Medium
• Last Updated : 01 Jun, 2021

Given n queries of the form range [L, R]. The task is to find the maximum difference between two prime numbers in the range for each query. If there are no prime in the range then print 0. All ranges are below 100005.

Examples:

```Input : Q = 3
query1 = [2, 5]
query2 = [2, 2]
query3 = [24, 28]
Output : 3
0
0
In first query, 2 and 5 are prime number
in the range with maximum difference which
is 3. In second there
is only 1 prime number in range, so output
is 0. And in third query, there is no prime number in the given range so the output is 0.```

The idea is to compute Prime numbers using Sieve of Eratosthenes along with some precomputing.

Below are the step to solve the question:
Step 1: Find the prime numbers using Sieve of Eratosthenes algorithm.
Step 2: Make an array, let say prefix[], where prefix[i] represents largest prime number smaller or equal to i.
Step 3: Make an array, let say suffix[], where suffix[i] represents smallest prime number greater or equal to i.
Step 4: Now for each query having [L, R], do the following:

```  if (prefix[R]  R)
return 0;
else
return prefix[R] - suffix[L];```

Below is the implementation of this approach:

## C++

 `// CPP program to find maximum differences between``// two prime numbers in given ranges``#include ``using` `namespace` `std;``#define MAX 100005` `// Declare global variables to assign heap memory and avoid``// stack overflow``bool` `prime[MAX];``int` `prefix[MAX], suffix[MAX];` `// Precompute Sieve, Prefix array, Suffix array``void` `precompute(``int` `prefix[], ``int` `suffix[])``{``    ``memset``(prime, ``true``, ``sizeof``(prime));` `    ``// Sieve of Eratosthenes``    ``for` `(``int` `i = 2; i * i < MAX; i++) {``        ``if` `(prime[i]) {``            ``for` `(``int` `j = i * i; j < MAX; j += i)``                ``prime[j] = ``false``;``        ``}``    ``}` `    ``prefix[1] = 1;``    ``suffix[MAX - 1] = 1e9 + 7;` `    ``// Precomputing Prefix array.``    ``for` `(``int` `i = 2; i < MAX; i++) {``        ``if` `(prime[i])``            ``prefix[i] = i;``        ``else``            ``prefix[i] = prefix[i - 1];``    ``}` `    ``// Precompute Suffix array.``    ``for` `(``int` `i = MAX - 1; i > 1; i--) {``        ``if` `(prime[i])``            ``suffix[i] = i;``        ``else``            ``suffix[i] = suffix[i + 1];``    ``}``}` `// Function to solve each query``int` `query(``int` `prefix[], ``int` `suffix[], ``int` `L, ``int` `R)``{``    ``if` `(prefix[R] < L || suffix[L] > R)``        ``return` `0;``    ``else``        ``return` `prefix[R] - suffix[L];``}` `// Driven Program``int` `main()``{``    ``int` `q = 3;``    ``int` `L[] = { 2, 2, 24 };``    ``int` `R[] = { 5, 2, 28 };` `    ``precompute(prefix, suffix);` `    ``for` `(``int` `i = 0; i < q; i++)``        ``cout << query(prefix, suffix, L[i], R[i]) << endl;` `    ``return` `0;``}`

## Java

 `// Java program to find maximum differences between``// two prime numbers in given ranges` `public` `class` `GFG {` `    ``final` `static` `int` `MAX = ``100005``;` `    ``// Precompute Sieve, Prefix array, Suffix array``    ``static` `void` `precompute(``int` `prefix[], ``int` `suffix[])``    ``{``        ``boolean` `prime[] = ``new` `boolean``[MAX];``        ``for` `(``int` `i = ``0``; i < MAX; i++) {``            ``prime[i] = ``true``;``        ``}` `        ``// Sieve of Eratosthenes``        ``for` `(``int` `i = ``2``; i * i < MAX; i++) {``            ``if` `(prime[i]) {``                ``for` `(``int` `j = i * i; j < MAX; j += i) {``                    ``prime[j] = ``false``;``                ``}``            ``}``        ``}` `        ``prefix[``1``] = ``1``;``        ``suffix[MAX - ``1``] = (``int``)1e9 + ``7``;` `        ``// Precomputing Prefix array.``        ``for` `(``int` `i = ``2``; i < MAX; i++) {``            ``if` `(prime[i]) {``                ``prefix[i] = i;``            ``}``            ``else` `{``                ``prefix[i] = prefix[i - ``1``];``            ``}``        ``}` `        ``// Precompute Suffix array.``        ``for` `(``int` `i = MAX - ``2``; i > ``1``; i--) {``            ``if` `(prime[i]) {``                ``suffix[i] = i;``            ``}``            ``else` `{``                ``suffix[i] = suffix[i + ``1``];``            ``}``        ``}``    ``}` `    ``// Function to solve each query``    ``static` `int` `query(``int` `prefix[], ``int` `suffix[], ``int` `L,``                     ``int` `R)``    ``{``        ``if` `(prefix[R] < L || suffix[L] > R) {``            ``return` `0``;``        ``}``        ``else` `{``            ``return` `prefix[R] - suffix[L];``        ``}``    ``}` `    ``// Driven Program``    ``public` `static` `void` `main(String[] args)``    ``{``        ``int` `q = ``3``;``        ``int` `L[] = { ``2``, ``2``, ``24` `};``        ``int` `R[] = { ``5``, ``2``, ``28` `};` `        ``int` `prefix[] = ``new` `int``[MAX], suffix[]``                                     ``= ``new` `int``[MAX];``        ``precompute(prefix, suffix);` `        ``for` `(``int` `i = ``0``; i < q; i++) {``            ``System.out.println(``                ``query(prefix, suffix, L[i], R[i]));``        ``}``    ``}``}``/*This code is contributed by Rajput-Ji*/`

## Python3

 `# Python 3 program to find maximum``# differences between two prime numbers``# in given ranges``from` `math ``import` `sqrt` `MAX` `=` `100005` `# Precompute Sieve, Prefix array, Suffix array``def` `precompute(prefix, suffix):``    ``prime ``=` `[``True` `for` `i ``in` `range``(``MAX``)]` `    ``# Sieve of Eratosthenes``    ``k ``=` `int``(sqrt(``MAX``))``    ``for` `i ``in` `range``(``2``, k, ``1``):``        ``if` `(prime[i]):``            ``for` `j ``in` `range``(i ``*` `i, ``MAX``, i):``                ``prime[j] ``=` `False` `    ``prefix[``1``] ``=` `1``    ``suffix[``MAX` `-` `1``] ``=` `int``(``1e9` `+` `7``)` `    ``# Precomputing Prefix array.``    ``for` `i ``in` `range``(``2``, ``MAX``, ``1``):``        ``if` `(prime[i]):``            ``prefix[i] ``=` `i``        ``else``:``            ``prefix[i] ``=` `prefix[i ``-` `1``]` `    ``# Precompute Suffix array.``    ``i ``=` `MAX` `-` `2``    ``while``(i > ``1``):``        ``if` `(prime[i]):``            ``suffix[i] ``=` `i``        ``else``:``            ``suffix[i] ``=` `suffix[i ``+` `1``]``        ``i ``-``=` `1` `# Function to solve each query`  `def` `query(prefix, suffix, L, R):``    ``if` `(prefix[R] < L ``or` `suffix[L] > R):``        ``return` `0``    ``else``:``        ``return` `prefix[R] ``-` `suffix[L]`  `# Driver Code``if` `__name__ ``=``=` `'__main__'``:``    ``q ``=` `3``    ``L ``=` `[``2``, ``2``, ``24``]``    ``R ``=` `[``5``, ``2``, ``28``]` `    ``prefix ``=` `[``0` `for` `i ``in` `range``(``MAX``)]``    ``suffix ``=` `[``0` `for` `i ``in` `range``(``MAX``)]``    ``precompute(prefix, suffix)` `    ``for` `i ``in` `range``(``0``, q, ``1``):``        ``print``(query(prefix, suffix,``                    ``L[i], R[i]))` `# This code is contributed by``# Surendra_Gangwar`

## C#

 `// C# program to find maximum differences between``// two prime numbers in given ranges``using` `System;` `public` `class` `GFG {` `    ``static` `readonly` `int` `MAX = 100005;` `    ``// Precompute Sieve, Prefix array, Suffix array``    ``static` `void` `precompute(``int``[] prefix, ``int``[] suffix)``    ``{``        ``bool``[] prime = ``new` `bool``[MAX];``        ``for` `(``int` `i = 0; i < MAX; i++) {``            ``prime[i] = ``true``;``        ``}` `        ``// Sieve of Eratosthenes``        ``for` `(``int` `i = 2; i * i < MAX; i++) {``            ``if` `(prime[i]) {``                ``for` `(``int` `j = i * i; j < MAX; j += i) {``                    ``prime[j] = ``false``;``                ``}``            ``}``        ``}` `        ``prefix[1] = 1;``        ``suffix[MAX - 1] = (``int``)1e9 + 7;` `        ``// Precomputing Prefix array.``        ``for` `(``int` `i = 2; i < MAX; i++) {``            ``if` `(prime[i]) {``                ``prefix[i] = i;``            ``}``            ``else` `{``                ``prefix[i] = prefix[i - 1];``            ``}``        ``}` `        ``// Precompute Suffix array.``        ``for` `(``int` `i = MAX - 2; i > 1; i--) {``            ``if` `(prime[i]) {``                ``suffix[i] = i;``            ``}``            ``else` `{``                ``suffix[i] = suffix[i + 1];``            ``}``        ``}``    ``}` `    ``// Function to solve each query``    ``static` `int` `query(``int``[] prefix, ``int``[] suffix, ``int` `L,``                     ``int` `R)``    ``{``        ``if` `(prefix[R] < L || suffix[L] > R) {``            ``return` `0;``        ``}``        ``else` `{``            ``return` `prefix[R] - suffix[L];``        ``}``    ``}` `    ``// Driven Program``    ``public` `static` `void` `Main()``    ``{``        ``int` `q = 3;``        ``int``[] L = { 2, 2, 24 };``        ``int``[] R = { 5, 2, 28 };` `        ``int``[] prefix = ``new` `int``[MAX];``        ``int``[] suffix = ``new` `int``[MAX];``        ``precompute(prefix, suffix);` `        ``for` `(``int` `i = 0; i < q; i++) {``            ``Console.WriteLine(``                ``query(prefix, suffix, L[i], R[i]));``        ``}``    ``}``}` `/*This code is contributed by 29AjayKumar*/`

## PHP

 ` 1; ``\$i``--)``    ``{``        ``if` `(``\$prime``[``\$i``])``            ``\$suffix``[``\$i``] = ``\$i``;``        ``else``            ``\$suffix``[``\$i``] = ``\$suffix``[``\$i` `+ 1];``    ``}``}` `// Function to solve each query``function` `query(``\$prefix``, ``\$suffix``, ``\$L``, ``\$R``)``{``    ``if` `(``\$prefix``[``\$R``] < ``\$L` `|| ``\$suffix``[``\$L``] > ``\$R``)``        ``return` `0;``    ``else``        ``return` `\$prefix``[``\$R``] - ``\$suffix``[``\$L``];``}` `// Driver Code``\$q` `= 3;``\$L` `= ``array``( 2, 2, 24 );``\$R` `= ``array``( 5, 2, 28 );` `\$prefix` `= ``array_fill``(0, ``\$MAX` `+ 1, 0);``\$suffix` `= ``array_fill``(0, ``\$MAX` `+ 1, 0);``precompute(``\$prefix``, ``\$suffix``);` `for` `(``\$i` `= 0; ``\$i` `< ``\$q``; ``\$i``++)``    ``echo` `query(``\$prefix``, ``\$suffix``,``               ``\$L``[``\$i``], ``\$R``[``\$i``]) . ``"\n"``;` `// This code is contributed by mits``?>`

## Javascript

 ``

Output:

```3
0
0 ```

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