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Fermat Number in Java

In this section, we will discuss what is Fermat number and also create Java programs to check if the given number is a Fermat number or not. The Fermat number program frequently asked in Java coding interviews and academics.

Fermat Number

Fermat number first studied by Pierre de Fermat. A non-negative odd integer that is in the form of 22k +1 (where k>=0) is known as a Fermat number. It is denoted by Fn. It is OEIS integer sequence A000215.

Fermat Number in Java

It is difficult to generate Fermat numbers. Since they are used in cryptography and pseudo-random number generation. It also has some powerful mathematical properties.

Fermat Number Example

First few Fermat numbers are 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617.

First 10 Fermat numbers:

F₀ = 220 +1 = 3

F₁ = 221 +1 = 5

F₂ = 222 +1 = 17

F₃ = 223 +1 = 257

F₄ = 224 +1 = 65537

F5 = 225 +1=4294967297

F6 = 226 +1=18446744073709551617

F₇ = 227 +1 = 340282366920938463463374607431768211457

F₈ = 228 +1 = 115792089237316195423570985008687907853269984665640564039457584007913129639937

F₉ = 229 +1 =

134078079299425970995740249982058461274793658205923933777235614437217640300
73546976801874298166903427690031858186486050853753882811946569946433649006084097
It is conjectured that just the first 5 numbers in this sequence are primes. It is OEIS sequence A019434.

F₀ = 220 +1 = 3

F₁ = 221 +1 = 5

F₂ = 222 +1 = 17

F₃ = 223 +1 = 257

F₄ = 224 +1 = 65537

Steps to Find Fermat Number

  1. Read or initialize an integer n.
  2. Calculate 2n by using the Math.pow() method and store the result in a variable (p).
  3. Now, calculate 2p and store the result in a variable (res).
  4. Add 1 in the variable (res) store the result in a variable (f). It is a Fermat number.

Let's implement the above steps in a Java program.

Fermat Number Java Program

FermatNumberExample1.java

Output:

3.0
5.0
17.0
257.0
65537.0
4.294967297E9
1.8446744073709552E19
3.4028236692093846E38
1.157920892373162E77
1.3407807929942597E154
Infinity

FermatNumberExample.java

Output:

First 10 Fermat numbers:
F[0] = 3
F[1] = 5
F[2] = 17
F[3] = 257
F[4] = 65537
F[5] = 4294967297
F[6] = 18446744073709551617
F[7] = 340282366920938463463374607431768211457
F[8] = 115792089237316195423570985008687907853269984665640564039457584007913129639937
F[9] = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006084097

First 12 Fermat numbers factored:
F[0] = 3 (PRIME)
F[1] = 5 (PRIME)
F[2] = 17 (PRIME)
F[3] = 257 (PRIME)
F[4] = 65537 (PRIME)
    Pollard rho try factor 4294967297 elapsed time = 8 ms (factor = 641).
F[5] = 641 * 6700417
    Pollard rho try factor 18446744073709551617 elapsed time = 33 ms (factor = 274177).
F[6] = 274177 * 67280421310721





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