Problems & Puzzles: Puzzles

Puzzle 956. A trio of integers and six squares

Find trios of integers N3={n1, n2, n3}, n1<n2<n3, such that every one of the six integers, nj+ni and nj-ni, j>i, are squares.

Example*

N3={150568, 420968, 434657}

420968 + 150568 = 756^2
420968 150568 = 520^2
434657 + 420968 = 925^2
434657 420968 = 117^2
434657 + 150568 = 765^2
434657 150568 = 533^2

Q1. Send three more solutions
Q2. Send your best quartet.
Q3.
Is it possible to get a trio solution using only primes?
_____
* From Math Notes, in Futility Closet

 

Contributions came from Emmanuel Vantieghem, Shyam Sunder Gupta

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On June 14, 2019, Emmanuel wrote:

The trio with smallest  a  is  {88642, 891458, 3713858}.
When I restricted myself to the cases in which  GCD(a,b,c) = 1, I found the following ones with  a < 10000000 :
{418304, 488000, 733025}
{577448, 1283048, 14438177}
{589568, 1782032, 5320193}
{1873432, 2288168, 2399057}
{2465208, 4504392, 7640833}
{8626688, 8845712, 9004913}
 
It is impossible to find trios with primes for, two odd numbers  x, y  such that  x - y  and  x+y  both are squares cannot exist.
 
I found three solutions such that  a+b+c  is prime : 
{150568, 420968, 434657}
{2465208, 4504392, 7640833}  and
{31631800, 38124104, 57387425}.

***

On June 14, 2019, Shyam wrote:

In connection with above puzzle three more solutions are 
(418304, 488000, 733025), (856350, 949986, 993250) and (602272, 1683872, 1738628).
Due to time constraint I could not test for large ranges.

***

On June 18, 2019 Jeff Heleen wrote:

For puzzle 956, as well as the solutions given by others, I found a few more.
I used 3 nested search loops where Ni +/- Nj = k^2, and k <= 10000. The
remaining three calculated k may or may not be <=10000. Here they are sorted
by smallest of the trio.

354568          3565832         14855432
797778          8023122         33424722
1010158         15832658        27122258
1355112         3788712         3911913
1418272         14263328        59421728
1673216         1952000         2932100
1761858         2040642         2843458
2216050         22286450        92846450
2309792         5132192         57752708
2358272         7128128         21280772
2409088         6735488         6954512
3425400         3799944         3973000
3714958         24163442        24417458
3764200         10524200        10866425
3764736         4392000         6597225
5016718         25719218        53597618
5306112         16038288        47881737
5420448         15154848        15647652
6692864         7808000         11728400
7047432         8162568         11373832
7377832         20627432        21298193
7493728         9152672         9596228
7707150         8549874         8939250
9433088         28512512        85123088
9636352         26941952        27818048
9860832         18017568        30563332
10457600        12200000        18325625
12196008        34098408        35207217
13701600        15199776        15892000
15056800        42096800        43465700
15058944        17568000        26388900
15856722        18365778        25591122
16860888        20593512        21591513
18218728        50937128        52593497
20126386        66475250        77764850
20408402        53586002        68516498
20496896        23912000        35918225
21408750        23749650        24831250
21681792        60619392        62590608
22011022        23147378        25433522
22186872        40539528        68767497
25445992        71143592        73457033
26578008        34261992        65678017
26633678        29316722        40606322
26771456        31232000        46913600
28189728        32650272        45495328
29974912        36610688        38384912
30828600        34199496        35757000
33882624        39528000        59375025
34506752        35382848        36019652
36935800        39381896        42719825
41961150        46549314        48669250

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