Nth Term of a Fibonacci Series of Primes formed by concatenating pairs

Nth Term of a Fibonacci Series of Primes formed by concatenating pairs

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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.

Input:_ X = 1, Y = 10_

Output:_ 1053_

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 countth 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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cs - placements mathematical searching sorting fibonacci prime number sieve tcs-coding-questions tcs-interview-experience

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