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Given two integers **X** and **Y**, the task is to perform the following operations:

- Find all prime numbers in the range
**[X, Y]**. - Generate all numbers possible by combining every pair of primes in the given range.
- Find the prime numbers among all the possible numbers generated above. Calculate the count of primes among them, say
**N**. - Print the Nth term of a Fibonacci Series formed by having the smallest and largest primes from the above list as the first two terms of the series.

**Examples:**

_Input: _X = 2 Y = 40

_Output: _34

Explanation:

All primes in the range [X, Y] = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]

All possible numbers generated by concatenating each pair of prime = [23, 25, 27, 211, 213, 217, 219, 223, 229, 231, 32, 35, 37, 311, 313, 319, 323, 329, 331, 337, 52, 53, 57, 511, 513, 517, 519, 523, 529, 531, 537, 72, 73, 75, 711, 713, 717, 719, 723, 729, 731, 737, 112, 113, 115, 117, 1113, 1117, 1119, 1123, 1129, 1131, 1137, 132, 133, 135, 137, 1311, 1317, 1319, 1323, 1329, 1331, 1337, 172, 173, 175, 177, 1711, 1713, 1719, 1723, 1729, 1731, 1737, 192, 193, 195, 197, 1911, 1913, 1917, 1923, 1929, 1931, 1937, 232, 233, 235, 237, 2311, 2313, 2317, 2319, 2329, 2331, 2337, 292, 293, 295, 297, 2911, 2913, 2917, 2919, 2923, 2931, 2937, 312, 315, 317, 3111, 3113, 3117, 3119, 3123, 3129, 3137, 372, 373, 375, 377, 3711, 3713, 3717, 3719, 3723, 3729, 3731]

All primes among the generated numbers=[193, 3137, 197, 2311, 3719, 73, 137, 331, 523, 1931, 719, 337, 211, 23, 1117, 223, 1123, 229, 37, 293, 2917, 1319, 1129, 233, 173, 3119, 113, 53, 373, 311, 313, 1913, 1723, 317]

Count of the primes = 34

Smallest Prime = 23

Largest Prime = 3719

Therefore, the 34th term of the Fibonacci series having 23 and 3719 as the first two terms, is 13158006689.

_ X = 1, Y = 10_Input:

_ 1053_Output:

**Approach:**

Follow the steps below to solve the problem:

- Generate all possible primes using
**Sieve of Eratothenes**. - Traverse the range
**[X, Y]**and generate all primes in the range with the help of**primes[]**array generated in the step above. - Traverse the list of primes and generate all possible pairs from the list.
- For each pair, concatenate the two primes and check if their concatenation is a prime or not.
- Find the
**maximum**and**minimum**of all such primes and count all such primes obtained. - Finally, print the
**count**th of a Fibonacci series having**minimum**and**maximum**obtained in the above step as the first two terms of the series.

Below is the implementation of the above approach:

- Python

`## Python Program to implement`

`## the above approach`

`## Stores at each index if it's a `

`## prime or not`

`prime`

`**=**`

`[``True`

`**for**`

`i`

`**in**`

`range``(``100001``)]`

`## Sieve of Eratosthenes to `

`## generate all possible primes`

`**def**`

`SieveOfEratosthenes(): `

`p`

`**=**`

`2`

`**while**`

`(p`

`*****`

`p <``**=**`

`100000``):`

`## If p is a prime`

`**if**`

`(prime[p]`

`**==**`

`True``):`

`## Set all multiples of p as non-prime`

`**for**`

`i`

`**in**`

`range``(p`

`*****`

`p,`

`100001``, p):`

`prime[i]`

`**=**`

`False`

`p`

`**+=**`

`1`

`## Function to generate the `

`## required Fibonacci Series`

`**def**`

`fibonacciOfPrime(n1, n2):`

`SieveOfEratosthenes()`

`## Stores all primes between`

`## n1 and n2`

`initial`

`**=**`

`[]`

`## Generate all primes between`

`## n1 and n2`

`**for**`

`i`

`**in**`

`range``(n1, n2):`

`**if**`

`prime[i]:`

`initial.append(i)`

`## Stores all concatenations`

`## of each pair of primes`

`now`

`**=**`

`[]`

`## Generate all concatenations`

`## of each pair of primes`

`**for**`

`a`

`**in**`

`initial:`

`**for**`

`b`

`**in**`

`initial:`

`**if**`

`a !``**=**`

`b:`

`c`

`**=**`

`str``(a)`

`**+**`

`str``(b)`

`now.append(``int``(c))`

`## Stores the primes out of the`

`## numbers generated above`

`current`

`**=**`

`[]`

`**for**`

`x`

`**in**`

`now:`

`**if**`

`prime[x]:`

`current.append(x)`

`## Store the unique primes`

`current`

`**=**`

`set``(current)`

`## Find the minimum`

`first`

`**=**`

`min``(current)`

`## Find the minimum`

`second`

`**=**`

`max``(current)`

`## Find N`

`count`

`**=**`

`len``(current)`

`**-**`

`1`

`curr`

`**=**`

`1`

`**while**`

`curr < count:`

`c`

`**=**`

`first`

`**+**`

`second`

`first`

`**=**`

`second`

`second`

`**=**`

`c`

`curr`

`**+=**`

`1`

`## Print the N-th term`

`## of the Fibonacci Series`

`**print**``(c)`

`## Driver Code`

`**if**`

`__name__`

`**==**`

`"__main__"``:`

`x`

`**=**`

`2`

`y`

`**=**`

`40`

`fibonacciOfPrime(x, y)`

**Output:**

`13158006689`

** Time Complexity:**_ O(N2 + log(log(maxm))), where it takes O(N2) to generate all pairs and O(1) to check if a number is prime or not and maxm is the size of prime[]_

** Auxiliary Space:**_ O(maxm)_

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A Computer Science portal for geeks. It contains well written, well thought and well explained computer science and programming articles, quizzes and practice/competitive programming/company interview Questions.