2 minor update

JavaScript (ES7), 55 bytes

n=>(g=k=>k>0&&n%--k?g(k):k==1)(n=(49+n)**.5-7)|g(n+=14)


Try it online!

How?

Given a positive integer $$\n\$$, we're looking for a prime $$\x\$$ such that $$\x(x+14)=n\$$ or $$\x(x-14)=n\$$.

$$x^2+14x-n=0\tag{1}$$ $$x^2-14x-n=0\tag{2}$$

The positive root of $$\(1)\$$ is:

$$x_0=\sqrt{49+n}-7$$

and the positive root of $$\(2)\$$ is:

$$x_1=\sqrt{49+n}+7$$

Therefore, the problem is equivalent to testing whether either $$\x_0\$$ or $$\x_1\$$ is prime.

To do that, we use the classic recursive primality test function, with an additional test to make sure that it does not loop forever if it's given an irrational number as input.

g = k =>    // k = explicit input; this is the divisor
// we assume that the implicit input n is equal to k on the initial call
k > 0 &&  // abort if k is negative, which may happen if n is irrational
n % --k ? // decrement k; if k is not a divisor of n:
g(k)    //   do a recursive call
:         // else:
k == 1  //   returns true if k is equal to 1 (n is prime)
//   or false otherwise (n is either irrational or not a primecomposite integer)


Main wrapper function:

n => g(n = (49 + n) ** .5 - 7) | g(n += 14)


JavaScript (ES7), 55 bytes

n=>(g=k=>k>0&&n%--k?g(k):k==1)(n=(49+n)**.5-7)|g(n+=14)


Try it online!

How?

Given a positive integer $$\n\$$, we're looking for a prime $$\x\$$ such that $$\x(x+14)=n\$$ or $$\x(x-14)=n\$$.

$$x^2+14x-n=0\tag{1}$$ $$x^2-14x-n=0\tag{2}$$

The positive root of $$\(1)\$$ is:

$$x_0=\sqrt{49+n}-7$$

and the positive root of $$\(2)\$$ is:

$$x_1=\sqrt{49+n}+7$$

Therefore, the problem is equivalent to testing whether either $$\x_0\$$ or $$\x_1\$$ is prime.

To do that, we use the classic recursive primality test function, with an additional test to make sure that it does not loop forever if it's given an irrational number as input.

g = k =>    // k = explicit input; this is the divisor
// we assume that the implicit input n is equal to k on the initial call
k > 0 &&  // abort if k is negative, which may happen if n is irrational
n % --k ? // decrement k; if k is not a divisor of n:
g(k)    //   do a recursive call
:         // else:
k == 1  //   returns true if k is equal to 1 (n is prime)
//   or false otherwise (n is either irrational or not a prime integer)


Main wrapper function:

n => g(n = (49 + n) ** .5 - 7) | g(n += 14)


JavaScript (ES7), 55 bytes

n=>(g=k=>k>0&&n%--k?g(k):k==1)(n=(49+n)**.5-7)|g(n+=14)


Try it online!

How?

Given a positive integer $$\n\$$, we're looking for a prime $$\x\$$ such that $$\x(x+14)=n\$$ or $$\x(x-14)=n\$$.

$$x^2+14x-n=0\tag{1}$$ $$x^2-14x-n=0\tag{2}$$

The positive root of $$\(1)\$$ is:

$$x_0=\sqrt{49+n}-7$$

and the positive root of $$\(2)\$$ is:

$$x_1=\sqrt{49+n}+7$$

Therefore, the problem is equivalent to testing whether either $$\x_0\$$ or $$\x_1\$$ is prime.

To do that, we use the classic recursive primality test function, with an additional test to make sure that it does not loop forever if it's given an irrational number as input.

g = k =>    // k = explicit input; this is the divisor
// we assume that the implicit input n is equal to k on the initial call
k > 0 &&  // abort if k is negative, which may happen if n is irrational
n % --k ? // decrement k; if k is not a divisor of n:
g(k)    //   do a recursive call
:         // else:
k == 1  //   returns true if k is equal to 1 (n is prime)
//   or false otherwise (n is either irrational or a composite integer)


Main wrapper function:

n => g(n = (49 + n) ** .5 - 7) | g(n += 14)

1

JavaScript (ES7), 55 bytes

n=>(g=k=>k>0&&n%--k?g(k):k==1)(n=(49+n)**.5-7)|g(n+=14)


Try it online!

How?

Given a positive integer $$\n\$$, we're looking for a prime $$\x\$$ such that $$\x(x+14)=n\$$ or $$\x(x-14)=n\$$.

$$x^2+14x-n=0\tag{1}$$ $$x^2-14x-n=0\tag{2}$$

The positive root of $$\(1)\$$ is:

$$x_0=\sqrt{49+n}-7$$

and the positive root of $$\(2)\$$ is:

$$x_1=\sqrt{49+n}+7$$

Therefore, the problem is equivalent to testing whether either $$\x_0\$$ or $$\x_1\$$ is prime.

To do that, we use the classic recursive primality test function, with an additional test to make sure that it does not loop forever if it's given an irrational number as input.

g = k =>    // k = explicit input; this is the divisor
// we assume that the implicit input n is equal to k on the initial call
k > 0 &&  // abort if k is negative, which may happen if n is irrational
n % --k ? // decrement k; if k is not a divisor of n:
g(k)    //   do a recursive call
:         // else:
k == 1  //   returns true if k is equal to 1 (n is prime)
//   or false otherwise (n is either irrational or not a prime integer)


Main wrapper function:

n => g(n = (49 + n) ** .5 - 7) | g(n += 14)