Consider five positive integers a, b, c, d, and p, with gcd(a,b,c)=1, that satisfy this system of equations:
- a, b, and c are different positive integers.
- a2+b2+c2=d2
- a+b+c=p2
Determine the four smallest values of a+b+c+d+p.
*** Computer program/excel solver assisted solutions are welcome, but a semi-analytic (p&p and hand calculator) methodology is preferred.
clearvars,clc
sols=double.empty(0,6);
for a=1:500
for b=a+1:501
ab2=a^2+b^2;
for c=b+1:502
if gcd(gcd(a,b),c)==1
p2=a+b+c;
if round(sqrt(p2))^2==p2
d2=ab2+c^2;
if round(sqrt(d2))^2==d2
fprintf('%5d %5d %5d %5d %5d %7d
',...
[a b c sqrt(d2) sqrt(p2)], sum([a b c sqrt(d2) sqrt(p2)]))
sols(end+1,:)=[a b c sqrt(d2) sqrt(p2) sum([a b c sqrt(d2) sqrt(p2)])];
end
end
end
end
end
end
sols=sortrows(sols,6);
sols(1:10,:)
produces, at the end, the ten lowest sums:
a b c d p sum
1 6 18 19 5 49
12 16 21 29 7 85
6 21 22 31 7 87
4 17 28 33 7 89
1 18 30 35 7 91
6 10 33 35 7 91
30 45 46 71 11 203
28 36 57 73 11 205
25 34 62 75 11 207
18 34 69 79 11 211
Of course a, b, and c can be permuted in each case.
The first four satisfy the requirements.
All the solutions up to a total of 500 are (showing there's a tie for 10th place):
sols =
1 6 18 19 5 49
12 16 21 29 7 85
6 21 22 31 7 87
4 17 28 33 7 89
1 18 30 35 7 91
6 10 33 35 7 91
30 45 46 71 11 203
28 36 57 73 11 205
25 34 62 75 11 207
18 34 69 79 11 211
21 30 70 79 11 211
8 49 64 81 11 213
4 45 72 85 11 217
13 26 82 87 11 219
9 28 84 89 11 221
6 13 102 103 11 235
49 50 70 99 13 281
36 61 72 101 13 283
25 68 76 105 13 287
40 41 88 105 13 287
36 37 96 109 13 291
14 70 85 111 13 293
30 33 106 115 13 297
4 77 88 117 13 299
28 32 109 117 13 299
6 61 102 119 13 301
4 60 105 121 13 303
21 28 120 125 13 307
93 94 102 167 17 473
69 88 132 173 17 479
72 84 133 173 17 479
66 85 138 175 17 481
49 112 128 177 17 483
54 90 145 179 17 485
45 100 144 181 17 487
38 109 142 183 17 489
58 74 157 183 17 489
36 105 148 185 17 491
40 96 153 185 17 491
40 85 164 189 17 495
21 126 142 191 17 497
34 90 165 191 17 497
. . .
Edited on August 11, 2023, 9:03 am
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Posted by Charlie
on 2023-08-11 09:01:04 |
computer solution
