Find two Pythagorean triples A,B,C and D,E,F such that A+D is a triangular number, B+E is a triangular number, and C+F is a triangular number.
Triangular numbers are numbers of the form N(N+1)/2, such as 1, 3, 6, 10, 15, ... .
The program builds a table of Pythagorean triples and then looks for pairs to add together to get three triangular numbers:
clearvars,clc
Pyth=double.empty(0,3);
for m=1:500
for n=1:m-1
if gcd(m,n)==1
if mod(m*n,2)==0
a=m^2-n^2;
b=2*m*n;
c=m^2+n^2;
for k=1:300
p=k*[a b c];
if a<300
Pyth(end+1,:)=p;
else
break
end
end
end
end
end
end
for a=1:length(Pyth)
p1=Pyth(a,:);
for b=p:length(Pyth);
p2=Pyth(b,:);
if isTri(p1(1)+p2(1))
if isTri(p1(2)+p2(2))
if isTri(p1(3)+p2(3))
fprintf('%6d %6d %6d\n',p1)
fprintf('%6d %6d %6d\n',p2)
fprintf('%6d %6d %6d \n',p1+p2)
for i=1:3
fprintf('%6d ', whichTri(p1(i)+p2(i)))
end
fprintf('\n')
disp(' ')
end
end
end
end
end
function t=isTri(n)
sq=2*n;
sr=round(sqrt(sq));
if sr*(sr+1)/2==n
t=true;
else
t=false;
end
end
function w=whichTri(n)
sq=2*n;
sr=round(sqrt(sq));
if sr*(sr+1)/2==n
w=sr;
else
w=0;
end
end
The first instance of valid pairs of Pythagorean triples is
39, 52, 65 and 132, 224, 260
Adding corresponding elements gives
171, 276, 325
which are, respectively, the 18th, 23rd and 25th triangular numbers.
First two lines are the two Pythagorean Triples.
Third line is their sum.
Fourth line identifies which triangular number each is.
39 52 65
132 224 260
171 276 325
18 23 25
45 60 75
990 5400 5490
1035 5460 5565
45 104 105
45 60 75
10395 7200 12645
10440 7260 12720
144 120 159
66 88 110
234 1040 1066
300 1128 1176
24 47 48
168 224 280
183 1856 1865
351 2080 2145
26 64 65
183 244 305
520 576 776
703 820 1081
37 40 46
189 252 315
1242 1344 1830
1431 1596 2145
53 56 65
195 260 325
7065 4396 8321
7260 4656 8646
120 96 131
225 300 375
3780 1296 3996
4005 1596 4371
89 56 93
330 440 550
1155 1100 1595
1485 1540 2145
54 55 65
342 456 570
1254 720 1446
1596 1176 2016
56 48 63
426 568 710
900 560 1060
1326 1128 1770
51 47 59
630 840 1050
1323 336 1365
1953 1176 2415
62 48 69
630 840 1050
855 24360 24375
1485 25200 25425
54 224 225
720 960 1200
1050 3600 3750
1770 4560 4950
59 95 99
750 1000 1250
1020 128 1028
1770 1128 2278
59 47 67
771 1028 1285
2310 1600 2810
3081 2628 4095
78 72 90
175 420 455
420 400 580
595 820 1035
34 40 45
285 684 741
891 912 1275
1176 1596 2016
48 56 63
300 720 780
25806 20808 33150
26106 21528 33930
228 207 260
465 1116 1209
11781 150960 151419
12246 152076 152628
156 551 552
610 1464 1586
1805 3192 3667
2415 4656 5253
69 96 102
620 1488 1612
605 528 803
1225 2016 2415
49 63 69
675 1620 1755
4275 7560 8685
4950 9180 10440
99 135 144
675 1620 1755
1881 1540 2431
2556 3160 4186
71 79 91
715 1716 1859
3380 3744 5044
4095 5460 6903
90 104 117
Operation terminated by user during pythagoreanTriplesNo1
>>
|
|
Posted by Charlie
on 2024-01-31 13:57:22 |
computer solution
