clc, clearvars
pSet=primes(nthprime(55));
p5Set=nchoosek(pSet,5);
for wh5=1:length(p5Set)
p5=p5Set(wh5,:);
num0=prod(p5);
den=sum(p5.^2);
p2Set=nchoosek(p5,2);
for wh2=1:length(p2Set)
num=num0/prod(p2Set(wh2,:));
if num==den
disp(p5Set(wh5,:))
disp([p2Set(wh2,:) prod(p2Set(wh2,:))])
disp(p5.^2)
disp([num den])
disp(' ')
end
end
end
In the groups below, where the numbers satisfy the equation, the first row contains p(1) through p(5). The second row contains p(x), p(y) and their product p(x)*p(y). The third row contains the squares of the primes in the first row, which of course add up to the denominator of the fraction. The last row are, first the product of the five primes divided by the product of the two chosen to have the x and y subscripts, and second, the sum of the squares. These are of course equal, as each side of the equation is divided by the RHS of the original equation.
Primes included those through the 55th, which is 257.
The lowest product p(x)*p(y) found is 57 = 3 * 19, chosen from the five primes in the numerator, 3 5 11 19 43.
As the numbers tend to get higher as higher primes are allowed, it would seem that 57 is the lowest, and 69 the second lowest. This is almost certainly the case.
It's to be noticed that when expressed as p(a)*p(b)*p(c) / (p(a)^2+p(b)^2+p(c)^2+p(x)^2+p(y)^2) = 1, or p(a)*p(b)*p(c) = p(a)^2+p(b)^2+p(c)^2+p(x)^2+p(y)^2, that the LHS increases, generally as the cube of the size of the primes and the RHS as the square. This might be of use in an analytic proof.
Also note that the two primes subscripted by x and y are all the highest two (suscripts 4 and 5) of the set of five. If this is proven to be always the case, then the numbers 57 and 69 are varified.
3 5 11 19 23
3 23 69
9 25 121 361 529
1045 1045
3 5 11 19 43
3 19 57
9 25 121 361 1849
2365 2365
3 5 41 73 89
3 89 267
9 25 1681 5329 7921
14965 14965
3 5 67 131 149
3 149 447
9 25 4489 17161 22201
43885 43885
3 5 109 173 229
3 229 687
9 25 11881 29929 52441
94285 94285
3 5 131 163 251
3 251 753
9 25 17161 26569 63001
106765 106765
3 7 11 17 29
3 29 87
9 49 121 289 841
1309 1309
3 7 41 109 227
3 109 327
9 49 1681 11881 51529
65149 65149
3 17 19 23 79
3 79 237
9 289 361 529 6241
7429 7429
3 23 41 43 191
3 191 573
9 529 1681 1849 36481
40549 40549
5 7 19 59 89
5 59 295
25 49 361 3481 7921
11837 11837
5 7 43 113 139
5 139 695
25 49 1849 12769 19321
34013 34013
5 7 59 109 227
7 109 763
25 49 3481 11881 51529
66965 66965
5 7 79 179 223
7 179 1253
25 49 6241 32041 49729
88085 88085
5 11 17 37 41
11 37 407
25 121 289 1369 1681
3485 3485
5 11 19 37 67
11 37 407
25 121 361 1369 4489
6365 6365
5 11 41 47 131
5 131 655
25 121 1681 2209 17161
21197 21197
5 11 43 67 89
11 89 979
25 121 1849 4489 7921
14405 14405
5 13 19 41 43
13 41 533
25 169 361 1681 1849
4085 4085
5 13 61 139 157
13 139 1807
25 169 3721 19321 24649
47885 47885
5 17 23 31 41
23 31 713
25 289 529 961 1681
3485 3485
5 17 29 53 61
17 61 1037
25 289 841 2809 3721
7685 7685
5 19 23 31 67
23 31 713
25 361 529 961 4489
6365 6365
5 19 23 47 71
19 47 893
25 361 529 2209 5041
8165 8165
5 19 31 47 127
19 47 893
25 361 961 2209 16129
19685 19685
5 19 31 113 229
5 229 1145
25 361 961 12769 52441
66557 66557
5 19 43 79 163
19 79 1501
25 361 1849 6241 26569
35045 35045
5 19 53 101 197
19 101 1919
25 361 2809 10201 38809
52205 52205
5 19 83 131 173
19 173 3287
25 361 6889 17161 29929
54365 54365
5 19 107 227 241
19 241 4579
25 361 11449 51529 58081
121445 121445
5 23 29 53 59
23 59 1357
25 529 841 2809 3481
7685 7685
5 23 31 41 47
31 41 1271
25 529 961 1681 2209
5405 5405
5 23 31 67 79
23 67 1541
25 529 961 4489 6241
12245 12245
5 23 67 139 149
23 149 3427
25 529 4489 19321 22201
46565 46565
5 29 37 47 101
37 47 1739
25 841 1369 2209 10201
14645 14645
5 29 37 61 73
29 73 2117
25 841 1369 3721 5329
11285 11285
5 29 47 73 191
29 73 2117
25 841 2209 5329 36481
44885 44885
5 31 47 53 79
47 53 2491
25 961 2209 2809 6241
12245 12245
5 31 53 101 113
31 113 3503
25 961 2809 10201 12769
26765 26765
5 37 41 61 67
41 67 2747
25 1369 1681 3721 4489
11285 11285
5 37 41 79 137
37 79 2923
25 1369 1681 6241 18769
28085 28085
5 37 43 71 109
43 71 3053
25 1369 1849 5041 11881
20165 20165
5 41 47 67 191
41 67 2747
25 1681 2209 4489 36481
44885 44885
5 41 53 113 149
41 113 4633
25 1681 2809 12769 22201
39485 39485
5 41 71 157 167
41 157 6437
25 1681 5041 24649 27889
59285 59285
5 41 89 113 167
41 167 6847
25 1681 7921 12769 27889
50285 50285
5 43 53 101 109
43 109 4687
25 1849 2809 10201 11881
26765 26765
5 43 59 127 131
43 127 5461
25 1849 3481 16129 17161
38645 38645
5 43 89 199 233
43 199 8557
25 1849 7921 39601 54289
103685 103685
5 43 107 131 199
43 199 8557
25 1849 11449 17161 39601
70085 70085
5 47 59 73 251
47 73 3431
25 2209 3481 5329 63001
74045 74045
5 47 59 83 109
47 109 5123
25 2209 3481 6889 11881
24485 24485
5 47 61 107 229
47 107 5029
25 2209 3721 11449 52441
69845 69845
5 47 83 179 251
47 179 8413
25 2209 6889 32041 63001
104165 104165
5 47 89 137 179
47 179 8413
25 2209 7921 18769 32041
60965 60965
5 53 71 97 149
71 97 6887
25 2809 5041 9409 22201
39485 39485
5 53 79 173 199
53 173 9169
25 2809 6241 29929 39601
78605 78605
5 59 67 113 179
67 113 7571
25 3481 4489 12769 32041
52805 52805
5 59 71 83 227
71 83 5893
25 3481 5041 6889 51529
66965 66965
5 59 73 97 131
59 131 7729
25 3481 5329 9409 17161
35405 35405
5 61 83 131 163
61 163 9943
25 3721 6889 17161 26569
54365 54365
5 61 83 179 227
61 179 10919
25 3721 6889 32041 51529
94205 94205
5 67 79 103 139
67 139 9313
25 4489 6241 10609 19321
40685 40685
5 67 101 107 211
101 107 10807
25 4489 10201 11449 44521
70685 70685
5 67 109 157 211
67 211 14137
25 4489 11881 24649 44521
85565 85565
5 67 127 151 229
67 229 15343
25 4489 16129 22801 52441
95885 95885
5 71 73 151 193
71 151 10721
25 5041 5329 22801 37249
70445 70445
5 71 79 103 137
71 137 9727
25 5041 6241 10609 18769
40685 40685
5 71 101 127 167
101 127 12827
25 5041 10201 16129 27889
59285 59285
5 79 103 127 131
103 131 13493
25 6241 10609 16129 17161
50165 50165
5 79 113 137 199
79 199 15721
25 6241 12769 18769 39601
77405 77405
5 79 113 139 223
113 139 15707
25 6241 12769 19321 49729
88085 88085
5 83 107 151 251
107 151 16157
25 6889 11449 22801 63001
104165 104165
5 89 97 179 233
97 179 17363
25 7921 9409 32041 54289
103685 103685
5 89 107 137 151
107 151 16157
25 7921 11449 18769 22801
60965 60965
5 97 107 131 179
97 179 17363
25 9409 11449 17161 32041
70085 70085
5 97 109 157 199
97 199 19303
25 9409 11881 24649 39601
85565 85565
5 97 151 163 241
151 163 24613
25 9409 22801 26569 58081
116885 116885
5 101 109 157 197
101 197 19897
25 10201 11881 24649 38809
85565 85565
5 101 139 167 173
139 167 23213
25 10201 19321 27889 29929
87365 87365
5 131 173 233 251
173 233 40309
25 17161 29929 54289 63001
164405 164405
5 157 163 181 241
163 241 39283
25 24649 26569 32761 58081
142085 142085
5 167 227 239 251
227 251 56977
25 27889 51529 57121 63001
199565 199565
5 173 197 211 239
211 239 50429
25 29929 38809 44521 57121
170405 170405
7 11 17 43 53
11 53 583
49 121 289 1849 2809
5117 5117
7 11 31 53 197
11 53 583
49 121 961 2809 38809
42749 42749
7 11 37 107 191
11 107 1177
49 121 1369 11449 36481
49469 49469
7 11 59 193 197
11 197 2167
49 121 3481 37249 38809
79709 79709
7 13 19 29 71
19 29 551
49 169 361 841 5041
6461 6461
7 13 19 41 113
13 41 533
49 169 361 1681 12769
15029 15029
7 13 47 79 199
7 199 1393
49 169 2209 6241 39601
48269 48269
7 17 23 37 61
17 61 1037
49 289 529 1369 3721
5957 5957
7 19 23 97 179
7 179 1253
49 361 529 9409 32041
42389 42389
7 19 31 79 173
19 79 1501
49 361 961 6241 29929
37541 37541
7 23 37 61 109
37 61 2257
49 529 1369 3721 11881
17549 17549
7 29 41 43 89
29 89 2581
49 841 1681 1849 7921
12341 12341
7 29 43 89 127
29 127 3683
49 841 1849 7921 16129
26789 26789
7 29 61 71 127
61 71 4331
49 841 3721 5041 16129
25781 25781
7 29 61 167 197
29 197 5713
49 841 3721 27889 38809
71309 71309
7 29 79 197 251
29 251 7279
49 841 6241 38809 63001
108941 108941
7 31 37 103 191
31 103 3193
49 961 1369 10609 36481
49469 49469
7 31 89 163 257
31 257 7967
49 961 7921 26569 66049
101549 101549
7 37 73 79 191
73 79 5767
49 1369 5329 6241 36481
49469 49469
7 37 73 107 191
37 191 7067
49 1369 5329 11449 36481
54677 54677
7 37 79 173 241
37 241 8917
49 1369 6241 29929 58081
95669 95669
7 41 43 61 71
61 71 4331
49 1681 1849 3721 5041
12341 12341
7 41 53 67 101
53 101 5353
49 1681 2809 4489 10201
19229 19229
7 47 59 61 103
59 103 6077
49 2209 3481 3721 10609
20069 20069
7 47 59 131 229
59 131 7729
49 2209 3481 17161 52441
75341 75341
7 53 59 127 157
59 157 9263
49 2809 3481 16129 24649
47117 47117
7 53 59 167 199
59 167 9853
49 2809 3481 27889 39601
73829 73829
7 53 73 127 151
73 151 11023
49 2809 5329 16129 22801
47117 47117
7 59 73 97 109
97 109 10573
49 3481 5329 9409 11881
30149 30149
7 59 83 179 193
83 179 14857
49 3481 6889 32041 37249
79709 79709
7 59 127 151 193
127 151 19177
49 3481 16129 22801 37249
79709 79709
7 61 83 167 181
83 181 15023
49 3721 6889 27889 32761
71309 71309
7 73 89 107 173
89 173 15397
49 5329 7921 11449 29929
54677 54677
7 73 89 131 191
89 191 16999
49 5329 7921 17161 36481
66941 66941
7 73 101 127 241
73 241 17593
49 5329 10201 16129 58081
89789 89789
7 79 89 173 227
89 227 20203
49 6241 7921 29929 51529
95669 95669
7 79 101 131 149
131 149 19519
49 6241 10201 17161 22201
55853 55853
7 79 139 197 211
139 211 29329
49 6241 19321 38809 44521
108941 108941
7 79 149 173 193
149 193 28757
49 6241 22201 29929 37249
95669 95669
7 101 109 127 227
109 227 24743
49 10201 11881 16129 51529
89789 89789
7 101 181 199 241
181 241 43621
49 10201 32761 39601 58081
140693 140693
7 101 199 223 241
199 241 47959
49 10201 39601 49729 58081
157661 157661
7 113 211 229 239
229 239 54731
49 12769 44521 52441 57121
166901 166901
11 13 17 43 61
17 61 1037
121 169 289 1849 3721
6149 6149
11 13 17 97 107
11 107 1177
121 169 289 9409 11449
21437 21437
11 13 23 43 59
23 59 1357
121 169 529 1849 3481
6149 6149
11 13 23 127 223
11 127 1397
121 169 529 16129 49729
66677 66677
11 13 29 37 107
11 107 1177
121 169 841 1369 11449
13949 13949
11 13 29 181 229
11 181 1991
121 169 841 32761 52441
86333 86333
11 13 37 197 233
11 233 2563
121 169 1369 38809 54289
94757 94757
11 17 23 61 233
17 61 1037
121 289 529 3721 54289
58949 58949
11 17 29 47 73
29 73 2117
121 289 841 2209 5329
8789 8789
11 17 37 73 197
11 197 2167
121 289 1369 5329 38809
45917 45917
11 17 41 47 67
41 67 2747
121 289 1681 2209 4489
8789 8789
11 17 41 53 179
11 179 1969
121 289 1681 2809 32041
36941 36941
11 19 29 37 71
29 71 2059
121 361 841 1369 5041
7733 7733
11 19 43 61 151
19 151 2869
121 361 1849 3721 22801
28853 28853
11 23 41 71 157
23 157 3611
121 529 1681 5041 24649
32021 32021
11 23 61 107 113
61 107 6527
121 529 3721 11449 12769
28589 28589
11 23 71 101 137
71 101 7171
121 529 5041 10201 18769
34661 346
|
|
Posted by Charlie
on 2022-01-13 08:10:38 |
computer solution
