Find a Rocco number

JavaScript (ES7), 55 bytes

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

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How?

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

Hence the following quadratic equations:

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

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)

05AB1E, 8 bytes

Returns \$1\$ if \$n\$ is a Rocco number, or \$0\$ otherwise.

fDŠ/α14å

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How?

Given a positive integer \$n\$, we test whether there exists a prime factor \$p\$ of \$n\$ such that:

$$\left|p-\frac{n}{p}\right|=14$$

Commented

fDŠ/α14å  # expects a positive integer n as input       e.g. 2655
f         # push the list of unique prime factors of n  -->  2655, [ 3, 5, 59 ]
 D        # duplicate it                                -->  2655, [ 3, 5, 59 ], [ 3, 5, 59 ]
  Š       # moves the input n between the two lists     -->  [ 3, 5, 59 ], 2655, [ 3, 5, 59 ]
   /      # divide n by each prime factor               -->  [ 3, 5, 59 ], [ 885, 531, 45 ]
    α     # compute the absolute differences
          # between both remaining lists                -->  [ 882, 526, 14 ]
     14å  # does 14 appear in there?                    -->  1

Perl 6, 45 28 bytes

((*+49)**.5+(7|-7)).is-prime

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Uses Arnauld's construction, that \$\sqrt{n+49}\pm7\$ must be prime for \$n\$ to be a Rocco number.

Explanation:

 (*+49)**.5                   # Is the sqrt of input+49
           +(7|-7)            # Plus or minus 7
(                 ).is-prime  # Prime?