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 Conga Primes (Posted on 2013-03-23)

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.

 See The Solution Submitted by broll No Rating

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 part 1a (spoiler) | Comment 3 of 11 |

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            5463  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

 Posted by Charlie on 2013-03-26 08:35:08

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