A prime number p is called special if there e | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given $p = p_1 + p_2 = p_3 - p_4$ where $p, p_1, p_2, p_3, p_4$ are primes.Case $I$: If all $p_1, p_2, p_3, p_4$ are odd,then $p_1 + p_2$ is even,

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(B) Given $p = p_1 + p_2 = p_3 - p_4$ where $p, p_1, p_2, p_3, p_4$ are primes.Case $I$: If all $p_1, p_2, p_3, p_4$ are odd,then $p_1 + p_2$ is even,which implies $p$ is even. Since $p$ is a prime,$p = 2$. However,$p_1 + p_2 = 2$ is impossible for primes $p_1, p_2 \ge 2$.Case $II$: At least one of $p_1, p_2$ must be $2$. Let $p_2 = 2$. Then $p = p_1 + 2$. Since $p$ and $p_1$ are primes,$p_1$ must be odd (unless $p_1=2$,then $p=4$ not prime). If $p_1$ is odd,$p$ is odd.From $p = p_3 - p_4$,we have $p_3 = p + p_4$. Since $p$ is odd,for $p_3$ to be prime,$p_4$ must be $2$ (if $p_4$ were odd,$p_3$ would be even and $>2$,hence not prime).Thus,$p = p_1 + 2$ and $p = p_3 - 2$,which means $p_3 = p + 2$ and $p_1 = p - 2$.We need $p-2, p, p+2$ to be primes. This is a prime triplet of the form $(n, n+2, n+4)$. The only such triplet is $(3, 5, 7)$.Therefore,$p = 5$ is the only solution.The number of special primes is $1$.

$A$ prime number $p$ is called special if there exist primes $p_1, p_2, p_3, p_4$ such that $p = p_1 + p_2 = p_3 - p_4$. The number of special primes is

Similar Questions

$\cos(2x + 7) = a(2 - \sin x)$ can have a real solution for

If $\alpha$ and $\beta$ are the roots of the equation $x^2 - 3x + 1 = 0$,then what is the equation whose roots are $\frac{1}{\alpha - 2}$ and $\frac{1}{\beta - 2}$?

If $\frac{k}{kx+3}+\frac{3}{3x-k}=\frac{12x+5}{(kx+3)(3x-k)}$ for all $x \in R - \{-\frac{3}{k}, \frac{k}{3}\}$,then both the roots of the equation $kx^2-7x+3=0$ are

$A$ value of $x$ satisfying $150 x \equiv 35 \pmod{31}$ is

The roots of the equation $(x-a)(x-a-1)+(x-a-1)(x-a-2)+(x-a)(x-a-2)=0$ for $a \in R$ are always:

Vedclass Test Series

Exam Paper Generator

Online Exam Module