A380729 - OEIS

%I #65 Mar 24 2025 19:02:07

%S 9,27,215,2035,20095,200287,2000851,20002663,200008317,2000025997,

%T 20000082213,200000259021,2000000817463,20000002584459,

%U 200000008167303,2000000025828219,20000000081661683,200000000258208463,2000000000816541333

%N 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 From _David A. Corneth_, Feb 01 2025: (Start)

%C 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 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 [10000005, 10000018, 10000098, 10005204, 20002663] is a Pythagorean 8-digit quintuple, so a(8) <= 20002663.

%C From _David Consiglio, Jr._, Mar 05 2025: (Start)

%C a(9) <= 200008317 [100000000, 100000008, 100000220, 100016405, 200008317];

%C a(10) <= 2000026127 [1000000000, 1000000004, 1000000457, 1000051792, 2000026127];

%C a(11) <= 20000082345 [10000000008, 10000000030, 10000001006, 10000163645, 20000082345]. (End)

%C 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 Martin Fuller, <a href="/A380729/a380729.cpp.txt">C++ program</a>

%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a380/A380729.java">Java program</a> (github)

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanQuadruple.html">Pythagorean Quadruple</a>.

%F From _Martin Fuller_, Mar 16 2025: (Start)

%F a(n) > 2*10^(n-1) + ((2/3)*10^(n-1))^0.5.

%F   n even: a(n) > 2*10^(n-1) + 10^(n/2-1) * 2.5819888...

%F   n odd:  a(n) > 2*10^(n-1) + 10^((n-1)/2-1) * 8.1649658...

%F See proof in the C++ program. (End)

%e Pythagorean n-digit quintuples in strictly increasing order:

%e   [2, 4, 5, 6, 9];

%e   [10, 12, 14, 17, 27];

%e   [100, 101, 110, 118, 215];

%e   [1000, 1005, 1008, 1056, 2035];

%e   [10005, 10006, 10008, 10170, 20095];

%e   [100000, 100005, 100038, 100530, 200287];

%e   [1000001, 1000010, 1000040, 1001650, 2000851];

%e   [10000005, 10000018, 10000098, 10005204, 20002663];

%e   [100000000, 100000008, 100000220, 100016405, 200008317];

%e   [1000000005, 1000000020, 1000000240, 1000051728, 2000025997];

%e   [10000000001, 10000000102, 10000000742, 10000163580, 20000082213];

%e   [100000000010, 100000000054, 100000001169, 100000516808, 200000259021];

%e   [1000000000005, 1000000000062, 1000000001382, 1000001633476, 2000000817463];

%e   [10000000000006, 10000000000050, 10000000003649, 10000005165212, 20000002584459];

%e   [100000000000037, 100000000000142, 100000000003326, 100000016331100, 200000008167303];

%e   [1000000000000041, 1000000000000150, 1000000000012304, 1000000051643942, 2000000025828219];

%e   [10000000000000018, 10000000000000210, 10000000000017809, 10000000163305328, 20000000081661683];

%e   [100000000000000146, 100000000000000309, 100000000000013904, 100000000516402566, 200000000258208463];

%e   [1000000000000000210, 1000000000000000482, 1000000000000066436, 1000000001633015537, 2000000000816541333]

%o (Java) // See Links.

%Y Cf. A096910, A379744.

%K nonn,base,more

%O 1,1

%A _Jean-Marc Rebert_, Jan 31 2025

%E a(5) corrected by _Jinyuan Wang_, Feb 25 2025

%E a(8)-a(9) confirmed by _Sean A. Irvine_, Mar 06 2025

%E a(10)-a(19) from _Martin Fuller_, Mar 16 2025