Is it possible for a number to be a mersenne prime...

We can show that no number satisfies all three conditions. Let’s break down the definitions and examine the candidates:

1. Mersenne Primes:
   A Mersenne prime is a prime number of the form $2^p - 1$ where $p$ is a prime. The smallest few Mersenne primes are:

   • $3 = 2^2 - 1$
   • $7 = 2^3 - 1$
   • $31 = 2^5 - 1$
   • $127 = 2^7 - 1$, and so on.

2. Fibonacci Primes:
   A Fibonacci prime is a prime that appears in the Fibonacci sequence:

   $$0,\, 1,\, 1,\, 2,\, 3,\, 5,\, 8,\, 13,\, 21,\, 34,\, 55,\, 89,\, 144,\, \dots$$

   The Fibonacci primes include:

   • $2$ (which is $F(3)$),
   • $3$ (which is $F(4)$),
   • $5$ (which is $F(5)$),
   • $13$ (which is $F(7)$), etc.

3. Happy Primes:
   A happy number is one that eventually reaches 1 when you repeatedly replace it by the sum of the squares of its digits. A happy prime is a prime number that is also happy. For example, $7$ is happy because:

   $$7 \to 4^2+9^2=49 \to 4^2+9^2=97 \to 9^2+7^2=130 \to 1^2+3^2+0^2=10 \to 1^2+0^2=1.$$

Now, for a number to be all three—a Mersenne prime, a Fibonacci prime, and a happy prime—it must belong to the intersection of these three sets.

• Mersenne and Fibonacci Intersection:
  Look at the known small Mersenne primes. Out of these, the only one that is also a Fibonacci number is 3. (For instance, $7$, $31$, and $127$ do not appear in the Fibonacci sequence.)

• Checking Happiness for 3:
  Now we must check if $3$ is a happy number.

  • Start with $3$:
    $3^2 = 9$
  • Then $9$:
    $9^2 = 81$
  • Then $81$:
    $8^2 + 1^2 = 64 + 1 = 65$
  • Then $65$:
    $6^2 + 5^2 = 36 + 25 = 61$
  • Then $61$:
    $6^2 + 1^2 = 36 + 1 = 37$
  • Then $37$:
    $3^2 + 7^2 = 9 + 49 = 58$
  • Then $58$:
    $5^2 + 8^2 = 25 + 64 = 89$
  • Then $89$:
    $8^2 + 9^2 = 64 + 81 = 145$
  • Then $145$:
    $1^2 + 4^2 + 5^2 = 1 + 16 + 25 = 42$
  • Then $42$:
    $4^2 + 2^2 = 16 + 4 = 20$
  • Then $20$:
    $2^2 + 0^2 = 4 + 0 = 4$
  • Then $4$:
    $4^2 = 16$
  • Then $16$:
    $1^2 + 6^2 = 1 + 36 = 37$

  At this point, the number $37$ reappears, which means the sequence has entered a cycle that does not include $1$. Thus, 3 is not a happy number.

Since the only candidate that was both a Mersenne and a Fibonacci prime (namely, $3$) fails to be happy, there is no number that is simultaneously a Mersenne prime, a Fibonacci prime, and a happy prime.

Answer: No, it is not possible.