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.

9, 27, 215, 2035, 20095, 200287, 2000851, 20002663, 200008317, 2000025997, 20000082213, 200000259021, 2000000817463, 20000002584459, 200000008167303, 2000000025828219, 20000000081661683, 200000000258208463, 2000000000816541333

Offset

1,1

Comments

From David A. Corneth, Feb 01 2025: (Start)

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).

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)

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

From David Consiglio, Jr., Mar 05 2025: (Start)

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

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

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

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

Links

Table of n, a(n) for n=1..19.

Martin Fuller, C++ program

Sean A. Irvine, Java program (github)

Eric Weisstein's World of Mathematics, Pythagorean Quadruple.

Formula

From Martin Fuller, Mar 16 2025: (Start)

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

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

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

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

Example

Pythagorean n-digit quintuples in strictly increasing order:
  [2, 4, 5, 6, 9];
  [10, 12, 14, 17, 27];
  [100, 101, 110, 118, 215];
  [1000, 1005, 1008, 1056, 2035];
  [10005, 10006, 10008, 10170, 20095];
  [100000, 100005, 100038, 100530, 200287];
  [1000001, 1000010, 1000040, 1001650, 2000851];
  [10000005, 10000018, 10000098, 10005204, 20002663];
  [100000000, 100000008, 100000220, 100016405, 200008317];
  [1000000005, 1000000020, 1000000240, 1000051728, 2000025997];
  [10000000001, 10000000102, 10000000742, 10000163580, 20000082213];
  [100000000010, 100000000054, 100000001169, 100000516808, 200000259021];
  [1000000000005, 1000000000062, 1000000001382, 1000001633476, 2000000817463];
  [10000000000006, 10000000000050, 10000000003649, 10000005165212, 20000002584459];
  [100000000000037, 100000000000142, 100000000003326, 100000016331100, 200000008167303];
  [1000000000000041, 1000000000000150, 1000000000012304, 1000000051643942, 2000000025828219];
  [10000000000000018, 10000000000000210, 10000000000017809, 10000000163305328, 20000000081661683];
  [100000000000000146, 100000000000000309, 100000000000013904, 100000000516402566, 200000000258208463];
  [1000000000000000210, 1000000000000000482, 1000000000000066436, 1000000001633015537, 2000000000816541333]

Prog

(Java) // See Links.

Crossrefs

Cf. A096910, A379744.

Sequence in context: A340235 A217640 A227474 * A286524 A057901 A320676

Adjacent sequences: A380726 A380727 A380728 * A380730 A380731 A380732

Keyword

nonn,base,more

Author

Jean-Marc Rebert, Jan 31 2025

Extensions

a(5) corrected by Jinyuan Wang, Feb 25 2025

a(8)-a(9) confirmed by Sean A. Irvine, Mar 06 2025

a(10)-a(19) from Martin Fuller, Mar 16 2025

Status

approved