[LGM]

Here is the code. Crudely done by brute force in a rather esoteric flavor of basic:

PRIMES = '2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61 ,67,71,73,79,83,89,97,101,103,109,113,127,131,137'
SWAP ',' WITH @AM IN PRIMES
ILIMIT = DCOUNT(PRIMES,@AM)
PRINT ILIMIT
FOR I = 4 TO ILIMIT
NBR = I
COMBOS = ''
COMBOX = ''
IDX = 1
FOR J = 1 TO I
FOR K = J+1 TO I
FOR L = K+1 TO I
GOSUB FIND.DUPLICATE
IF NOT(FOUND.DUPLICATE) THEN
COMBOS<-1> = PRIMES*PRIMES*PRIMES
COMBOX<-1> = J:',':K:',':L
END
NEXT L
NEXT K
NEXT J
GOSUB CHECK.SOLUTION
NEXT I
PRINT 'DONE'
STOP
*
SOLVED:*
PRINT '---------------------'
PRINT 'SOLVED'
PRINT NBR
FOR M = DCOUNT(COMBOX,@AM) TO 1 STEP -1
PRINT COMBOX
NEXT M
INPUT CH,1_
RETURN
*
CHECK.SOLUTION: *
FOR J = 1 TO I
FOR K = J + 1 TO I
NBR.MATCHES = 0
FOR M = DCOUNT(COMBOS,@AM) TO 1 STEP -1
A.COMBO = COMBOS
IF MOD(A.COMBO,PRIMES)=0 AND MOD(A.COMBO,PRIMES)=0 THEN
NBR.MATCHES = NBR.MATCHES + 1
IF NBR.MATCHES > 1 THEN RETURN
END
NEXT M
IF NBR.MATCHES = 0 THEN RETURN
NEXT K
NEXT J
GOSUB SOLVED
RETURN
*
FIND.DUPLICATE: *
FOUND.DUPLICATE = 0
FOR M = DCOUNT(COMBOS,@AM) TO 1 STEP -1
A.COMBO = COMBOS
J.MOD = MOD(A.COMBO,PRIMES)
K.MOD = MOD(A.COMBO,PRIMES)
L.MOD = MOD(A.COMBO,PRIMES)
IF J.MOD = 0 THEN
IF K.MOD=0 OR L.MOD=0 THEN
FOUND.DUPLICATE = 1
END
END ELSE IF K.MOD=0 AND L.MOD=0 THEN
FOUND.DUPLICATE = 1
END
NEXT M
RETURN