x^2-y^2 = y^2-z^2 = 5 is a classic problem that can be solved in the rationals, with, e.g.:

(⁴⁹/₁₂)²-(⁴¹/₁₂)² = (⁴¹/₁₂)²-(³¹/₁₂)² = 5 (Fibonacci).

We seek non-trivial rational solutions to x^2-y^2 = y^2-z^2 = P, with P prime. Since we can always find compound multiples of such solutions with other primes happily joining the chain, let's call these paragons 'conga primes'. (Conversely, primes that only appear in conjunction with other primes could be 'tango primes', since it takes at least two...)

1. Solve over the rationals:
x^2-y^2 = y^2-z^2 = 7
x^2-y^2 = y^2-z^2 = 41

2. Give an example of a 'conga prime', P, greater than 41, such that x^2-y^2 = y^2-z^2 = P.

DATA  2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47 , 53 , 59
DATA 61 , 67 , 71 , 73 , 79 , 83 , 89 , 97 , 101 , 103 , 107 , 109 , 113

CLS
DIM prm(30)
FOR i = 1 TO 30: READ prm(i): NEXT

FOR tot = 6 TO 999999
  FOR a = -INT(-tot / 3) TO tot - 3
   r = tot - a
   FOR b = -INT(-r / 2) TO r - 1
    IF b >= a THEN EXIT FOR
    c = r - b
    IF c < b THEN
      b2 = b * b
      diff2 = b2 - c * c
      IF a * a - b2 = diff2 THEN
        FOR p = 1 TO 30
          q = diff2 / prm(p)
          IF q = INT(q) THEN
            sr = INT(SQR(q) + .5)
            IF sr * sr = q THEN
               if gcd(gcd(a,b),c)=1 then
                 PRINT a; b; c, sr, prm(p)
               end if 
            END IF
          END IF
        NEXT
      END IF
    END IF
   NEXT
  NEXT
NEXT tot

FUNCTION gcd (x, y)
dnd = x: dvr = y
IF dnd < dvr THEN SWAP dnd, dvr
DO
    q = INT(dnd / dvr)
    r = dnd - q * dvr
    dnd = dvr: dvr = r
LOOP UNTIL r = 0
gcd = dnd
END FUNCTION

finds

49  41  31        12            5
463  337  113     120           7

meaning

(49/12)^2 - (41/12)^2 = (41/12)^2 - (31/12)^2 = 5

(463/120)^2 - (337/120)^2 = (337/120)^2 - (113/120)^2 = 7