A284477 Pairs of integers (x, y), such that x^2 + 1 and y^2 + 1, 1 < y < x, have the same distinct prime factors.

7, 3, 18, 8, 117, 43, 239, 5, 378, 132, 843, 377, 2207, 987, 2943, 73, 4443, 53, 4662, 1568, 6072, 5118, 8307, 743, 8708, 2112, 9872, 2738, 31561, 4929, 103682, 46368, 271443, 121393, 853932, 76378, 1021693, 91383, 3539232, 41218, 3699356, 473654

Offset: 1

Author

Michel Lagneau, Mar 27 2017

Keywords

• nonn

Comments

The sequence appears to thin out quite abruptly; however, by solving the Diophantine equation x^2 + 1 = p (y^2 + 1) for a suitable prime p and selecting the solutions (x, y) for which p divides y^2 + 1, it is easy to generate larger pairs, such as (423222288438379883442890018716361, 66096216900526495715353522199871). - Giovanni Resta, Mar 27 2017

A very interesting property: the sequence contains a subsequence of pairs (Lucas numbers L(i), Fibonacci numbers F(i)) for i = 4, 6, 14, 16, 24, 36, ... These pairs are (L(4), F(4)), (L(6), F(6)), (L(14), F(14)), (L(16), F(16)), (L(24), F(24)), (L(26), F(26)), (L(34), F(34)), ... = (7, 3), (18, 8), (843, 377), (2207, 987), (103682, 46368), (271443, 121393), (12752043, 5702887), ... It seems that {i} = A090773(n) (numbers that are congruent to {4, 6} mod 10). - Michel Lagneau, Mar 28 2017

This is because L(i)^2+1 = 5*(F(i)^2+1) for even i, and 5 | F(i)^2 + 1 for i== 3,4,6,7 (mod 10). In fact (L(i), F(i)) for i in A090773 are the solutions of the generalized Pell equation x^2 + 1 = 5 (y^2 + 1) for which 5 | y^2 + 1. - Robert Israel, Apr 10 2017

Examples

			The pair (843, 377) is in the sequence because the prime factors of 843^2 + 1 and 377^2 + 1 are 2, 5, 61 and 233.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..68 (terms with x < 1.5*10^8)

Crossrefs

Cf. A002496, A089122, A128428.

Programs

Maple

A:= NULL:
for x from 2 to 10^5 do
  P:= numtheory:-factorset(x^2+1);
  if not assigned(R[P]) then R[P]:= x
  else A:= A, op(map(t -> (x,t), [R[P]]));
       R[P]:= R[P],x
  fi
od:
A; # Robert Israel, Apr 10 2017

Mathematica

d[n_] := First /@ FactorInteger[n]; Flatten@ Reap[ Do[ dx = d[x^2+1]; Do[ If[ dx == d[y^2+1], Sow[{x, y}]], {y, x-1}], {x, 1, 10^4}]][[2, 1]]

PARI

upto(n) = {my(l = List(), res=List()); for(i=1, n, f = factor(i^2+1)[, 1]; listput(l, [f, i])); listsort(l); for(i=1, n-1, if(l[i][1]==l[i+1][1], listput(res, [l[i+1][2], l[i][2]]))); listsort(res); res} \\ David A. Corneth, Mar 28 2017

Extensions

a(29)-a(34) from Giovanni Resta, Mar 27 2017

a(35)-a(42) from David A. Corneth, Mar 28 2017