A096910 -id:A096910 - cp's OEIS frontend

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

Jean-Marc Rebert, Jan 31 2025

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

			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]

Martin Fuller, C++ program
Sean A. Irvine, Java program (github)
Eric Weisstein's World of Mathematics, Pythagorean Quadruple.

Cf. A096910, A379744.

Java
```
// See Links.
```

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)

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

A337251 Positive integers k such that k^2 = A^2+B^2+C^2 and A^3+B^3+C^3 = m^3, where gcd(A,B,C) = 1 and A, B, C, m are positive integers.

Original entry on oeis.org

75, 119, 551, 755, 4501, 4895, 16371, 56863, 61091, 74201, 201797, 336709, 534793, 596827, 879397, 1007541

Offset: 1

Mo Li, Aug 21 2020

From Chai Wah Wu, Sep 04 2020: (Start)
A. Martin and R. Davis showed that 91091088729334859 = sqrt(11868013975030087^2+16269106368215226^2+88837226814909894^2) is a term (see Links).
Table of values for k, A, B, C, m:
  k        A        B        C        m
  ---------------------------------------------
  75       14       23       70       71
  119      3        34       114      115
  551      18       349      426      493
  755      145      198      714      721
  4501     1016     2364     3693     4013
  4895     213      3450     3466     4357
  16371    3542     9286     13009    14497
  56863    6213     32194    46458    51157
  61091    29233    29574    44754    51985
  74201    32913    38444    54264    63185
  201797   106677   117252   124876   168373
  336709   110051   118044   295512   306467
  534793   116457   286752   436136   476393
  596827   202023   234550   510270   536023
  879397   43472    613560   628485   782597
  1007541  272267   417416   875656   914315
(End)

			56863 is in the sequence because 56863^2 = 6213^2 + 32194^2 + 46458^2, 6213^3 + 32194^3 + 46458^3 = 51157^3 and gcd(6213, 32194, 46458) = 1.

A. Martin and R. Davis, Solution of problem 143, Jahrbuch über die Fortschritte der Mathematik, Band 29, Jahrgang 1898, pub. 1900, p. 157.
Ed Pegg Jr.'s Math Puzzles, A^2 + B^2 + C^2 = Square, A^3 + B^3 + C^3 = Cube
Seiji Tomita, A simultaneous equation {x^2+y^2+z^2=u^2, x^3+y^3+z^3=v^3} has infinitely many integer solutions.

Cf. A096910.

cp's OEIS Frontend

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.

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