The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A094595

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A094595

Number of solutions to 1 == nxy (mod z) == nyz (mod x) == nzx (mod y) with 0 < x < y < z.
(history; published version)

#15 by OEIS Server at Thu Dec 21 11:00:59 EST 2023

LINKS

Robin Visser, <a href="/A094595/b094595_1.txt">Table of n, a(n) for n = 1..300</a>

#14 by Michel Marcus at Thu Dec 21 11:00:59 EST 2023

STATUS

reviewed

approved

Discussion

Thu Dec 21

11:00

OEIS Server: Installed first b-file as b094595.txt.

#13 by Joerg Arndt at Thu Dec 21 10:53:26 EST 2023

STATUS

proposed

reviewed

#12 by Stefano Spezia at Thu Dec 21 03:34:06 EST 2023

STATUS

editing

proposed

#11 by Stefano Spezia at Thu Dec 21 03:33:50 EST 2023

MATHEMATICA

Table[cnt=0; Do[d=Divisors[n*x*y-1]; Do[z=d[[i]]; If[z>y && Mod[n*x*z, y]==1 && Mod[n*y*z, x]==1, cnt++ ], {i, Length[d]}], {x, 3n-1}, {y, x+1, 2nx2n*x-1}]; cnt, {n, 64}]

STATUS

proposed

editing

Discussion

Thu Dec 21

03:34

Stefano Spezia: Now, Mma works

#10 by Robin Visser at Thu Dec 21 03:03:24 EST 2023

STATUS

editing

proposed

#9 by Robin Visser at Thu Dec 21 03:02:42 EST 2023

MATHEMATICA

Table[cnt=0; Do[d=Divisors[n*x*y-1]; Do[z=d[[i]]; If[z>y && Mod[n*x*z, y]==1 && Mod[n*y*z, x]==1, cnt++ ], {i, Length[d]}], {x, 2n+3n-1}, {y, x+1, 2n^2+2n2nx-1}]; cnt, {n, 64}]

STATUS

proposed

editing

Discussion

Thu Dec 21

03:03

Robin Visser: thanks, done!

#8 by Robin Visser at Thu Dec 21 02:44:43 EST 2023

STATUS

editing

proposed

Discussion

Thu Dec 21

02:55

Michel Marcus: so Mma should be amended to provide correct terms ?

#7 by Robin Visser at Mon Dec 18 22:36:27 EST 2023

COMMENTS

There are at least two solutions for all n > 1, given by (x,y,z) = (n+1, n^2+n+1, n^4+2n^3+2n^2+n-1) and (x,y,z) = (n+1, 2n^2+2n-1, 2n^2+2n+1). Following the linked solution by Silvia Fernández, it can be shown that all solutions satisfy 1 < x <= 3n - 1, x < y <= 2nx - 1, and y < z <= nxy - 1. Contrary to the above comment, there are solutions satisfying x > 2n+1. The first such example is given by (x,y,z) = (31,45,59) when n = 14. - Robin Visser, Dec 18 2023

LINKS

Robin Visser, <a href="/A094595/b094595_1.txt">Table of n, a(n) for n = 1..300</a>

Discussion

Thu Dec 21

02:27

Robin Visser: Added first 300 terms (computed with attached Sage code).

02:33

Robin Visser: Also corrected terms a(14), a(20), a(29), a(37), a(39) and a(58)  (before, they were 1 less than the correct value).  The given Mathematica code only checks for solutions satisfying x <= 2n+1, but in general there can be more solutions (Fernandez instead proves a correct upper bound of x <= 3n-1)!  E.g. the first three examples where x > 2n+1 are (x,y,z) = (31,45,59) for n = 14,  (49,57,83) for n = 20, and (62,69,259) for n = 29.

#6 by Robin Visser at Mon Dec 18 05:27:21 EST 2023

COMMENTS

All Following the linked solution by Silvia Fernández, it can be shown that all solutions satisfy 1 < x <= 3n - 1, x < y <= 2nx - 1, and y < z <= nxy - 1. Not all Contrary to the above comment, there are solutions satisfy satisfying x <= > 2n+1, the . The first counterexample example is given by (x,y,z) = (31,45,59) when n = 14. - Robin Visser, Dec 18 2023

LINKS

Donald Knuth, Silvia Fernández and Gerry Myerson, <a href="https://doi.org/10.2307/30037459">A Modular Triple: 11021</a>, Amer. Math. Monthly, 112 (2005), p. 279.