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A380729 Smallest n-digit number e such that there exists a primitive Pythagorean n-digit quintuple (a,b,c,d,e) with 10^(n-1) <= a < b < c < d < e < 10^n.

%I A380729 #65 Mar 24 2025 19:02:07
%S A380729 9,27,215,2035,20095,200287,2000851,20002663,200008317,2000025997,
%T A380729 20000082213,200000259021,2000000817463,20000002584459,
%U A380729 200000008167303,2000000025828219,20000000081661683,200000000258208463,2000000000816541333
%N A380729 Smallest n-digit number e such that there exists a primitive Pythagorean n-digit quintuple (a,b,c,d,e) with 10^(n-1) <= a < b < c < d < e < 10^n.
%C A380729 From _David A. Corneth_, Feb 01 2025: (Start)
%C A380729 Let s1, s2, s3, and s4 be primitive positive distinct integers such that s1^2 + s2^2 + s3^2 + s4^2 = S^2. As squares are 0 or 1 (mod 4) and the quintuple (s1, s2, s3, s4, S) is primitive they cannot all be even. Hence at least one of s1, s2, s3, s4 must be odd. Without loss of generality let s4 be odd. Then s1, s2 and s3 all have the same parity (even or odd).
%C A380729 We may write s1^2 + s2^2 + s3^2 + s4^2 = S^2 as s1^2 + s2^2 + s3^2 = S^2 - s4^2 = (S - s4)*(S + s4) and so look at divisor pairs of s1^2 + s2^2 + s3^2 that multiply to (S - s4)*(S + s4), solve for S and s4 to see if the quintuple (s1, s2, s3, s4, S) meets the criteria for a(n). (End)
%C A380729 [10000005, 10000018, 10000098, 10005204, 20002663] is a Pythagorean 8-digit quintuple, so a(8) <= 20002663.
%C A380729 From _David Consiglio, Jr._, Mar 05 2025: (Start)
%C A380729 a(9) <= 200008317 [100000000, 100000008, 100000220, 100016405, 200008317];
%C A380729 a(10) <= 2000026127 [1000000000, 1000000004, 1000000457, 1000051792, 2000026127];
%C A380729 a(11) <= 20000082345 [10000000008, 10000000030, 10000001006, 10000163645, 20000082345]. (End)
%C A380729 2e-(a+b+c+d) >= 1 for all quintuples, with equality if e is close to the lower bound. See C++ program for details. - _Martin Fuller_, Mar 18 2025
%H A380729 Martin Fuller, <a href="/A380729/a380729.cpp.txt">C++ program</a>
%H A380729 Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a380/A380729.java">Java program</a> (github)
%H A380729 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanQuadruple.html">Pythagorean Quadruple</a>.
%F A380729 From _Martin Fuller_, Mar 16 2025: (Start)
%F A380729 a(n) > 2*10^(n-1) + ((2/3)*10^(n-1))^0.5.
%F A380729   n even: a(n) > 2*10^(n-1) + 10^(n/2-1) * 2.5819888...
%F A380729   n odd:  a(n) > 2*10^(n-1) + 10^((n-1)/2-1) * 8.1649658...
%F A380729 See proof in the C++ program. (End)
%e A380729 Pythagorean n-digit quintuples in strictly increasing order:
%e A380729   [2, 4, 5, 6, 9];
%e A380729   [10, 12, 14, 17, 27];
%e A380729   [100, 101, 110, 118, 215];
%e A380729   [1000, 1005, 1008, 1056, 2035];
%e A380729   [10005, 10006, 10008, 10170, 20095];
%e A380729   [100000, 100005, 100038, 100530, 200287];
%e A380729   [1000001, 1000010, 1000040, 1001650, 2000851];
%e A380729   [10000005, 10000018, 10000098, 10005204, 20002663];
%e A380729   [100000000, 100000008, 100000220, 100016405, 200008317];
%e A380729   [1000000005, 1000000020, 1000000240, 1000051728, 2000025997];
%e A380729   [10000000001, 10000000102, 10000000742, 10000163580, 20000082213];
%e A380729   [100000000010, 100000000054, 100000001169, 100000516808, 200000259021];
%e A380729   [1000000000005, 1000000000062, 1000000001382, 1000001633476, 2000000817463];
%e A380729   [10000000000006, 10000000000050, 10000000003649, 10000005165212, 20000002584459];
%e A380729   [100000000000037, 100000000000142, 100000000003326, 100000016331100, 200000008167303];
%e A380729   [1000000000000041, 1000000000000150, 1000000000012304, 1000000051643942, 2000000025828219];
%e A380729   [10000000000000018, 10000000000000210, 10000000000017809, 10000000163305328, 20000000081661683];
%e A380729   [100000000000000146, 100000000000000309, 100000000000013904, 100000000516402566, 200000000258208463];
%e A380729   [1000000000000000210, 1000000000000000482, 1000000000000066436, 1000000001633015537, 2000000000816541333]
%o A380729 (Java) // See Links.
%Y A380729 Cf. A096910, A379744.
%K A380729 nonn,base,more
%O A380729 1,1
%A A380729 _Jean-Marc Rebert_, Jan 31 2025
%E A380729 a(5) corrected by _Jinyuan Wang_, Feb 25 2025
%E A380729 a(8)-a(9) confirmed by _Sean A. Irvine_, Mar 06 2025
%E A380729 a(10)-a(19) from _Martin Fuller_, Mar 16 2025