The twin primes 5 and 7are such that one half of their sum is a perfectnumber. A

Question

The twin primes 5 and 7are such that one half of their sum is a perfectnumber. Are there any other twin primes with this property?/ (Hint; Given the twin primes p and p+2, withp>5,1/2(p+p+2)=6k for some k>1.) Thank you in advance. The twin primes 5 and 7are such that one half of their sum is a perfectnumber. Are there any other twin primes with this property?/ (Hint; Given the twin primes p and p+2, withp>5,1/2(p+p+2)=6k for some k>1.) Thank you in advance.

Explanation / Answer

Suppose we have twin primes p, p + 2, p > 5, such that p + 1 =1/2 * (p + p + 2) is a perfect number. Since p + 1 is even,it must be of the form 2^(q - 1) * (2^q - 1), where 2^q - 1 is aMersenne prime. Now, since p and p + 2 are odd primes, neither is divisible by3. Thus, we must have p congruent to 2 mod 3 and p + 2congruent to 1 mod 3. (p congruent to 1 mod 3 makes p + 2congruent to 0 mod 3; and, of course, p is not 0 mod 3) Thusp + 1 is divisible by 3. Now 3 does not divide 2^(q -1). So 3 must divide the Mersenne prime, a contradiction. So there are no twin primes with this property.

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.