Unity
makes strength! However, what to unite and what to achieve?
Could we add prime numbers to get a square? But still have some order
and try to join (collect) consecutive prime numbers.
Naturally, we start with 2, which is clearly not a square, so we add 3
to it. We get 5, which is still not a square. We have no right to discourage,
we have only just begun!
2+3+5=10 It’s not a square…
2+3+5+7=17 It’s not a square…
2+3+5+7+11=28 It’s not a square…
2+3+5+7+11+13=41 It’s not a square…
2+3+5+7+11+13+17=58 It’s not a square…
2+3+5+7+11+13+17+19=77 It’s not a square…
Maybe it's time to give up? Let's have another try!
2+3+5+7+11+13+17+19+23=100 Unbelievable! Square!
Victory.
Sure, it was a harmless skit for fun when we're dealing with math, but
here in math, things don't end with surrender after a few failed attempts. A
true mathematician, if he finds no proof of the impossibility of solving the
problem, searches for the solution all his life.
The important thing is that we reached the solution relatively quickly
in this task. More precisely, to the first solution of countless many. Because
now we have to see if we can find a square if we start not from 2 but from 3.
Now we will be shorter, because the same game this time would be quite
boring: 26 consecutive prime numbers must be added to get a square.
It is interesting and encouraging that starting with the beautiful, what
a beautiful, wonderful number 5, with only four numbers we get a square:
5+7+11+13=36 Magnificent!
But after this wonderful case, a real horror began. After a while I
began to doubt that starting with 7 we would even be able to get to a square!
But imagine (and believe me) that collecting
1862 (one thousand eight hundred and sixty two)
consecutive primes (starting with 7), you get a square!
After this frenzied trial I felt confident in formulating (probably for
the first time) the epoch-making hypothesis that for every prime there exists a
corresponding number such that the sequence of primes beginning with the prime
in question and having a number equal to the corresponding number. Simply put,
the above problem always has a solution.
This hypothesis was supported (but not proven) by a small program (in
Python). The following table shows the "corresponding" number for the
first 24 prime numbers. (Over time it will be supplemented, but at the moment
the program is stuck on one new "week".)
2 –
9
3 –
23
5 –
4
7 –
1862
11
– 14
13
– 3
17
– 2
19
– 211
23
– 331
29
– 163
31
– 366
37
– 3
41
– 124
43
– 48
47
– 2
53
– 449
59
– 8403
61
– 121
67
– 35
71
– 2
73
– 4
79
– 105
83
– 77
89
– 43
Which is extremely interesting: how
"strangely" the corresponding numbers jump. There is apparently
incredible chaos with these prime numbers.
Another interesting thing, I would even say: it is
surprising that in quite a few cases it is enough to add the next one to the
prime number and the square is ready (corresponding number 2)! This feature
intrigued me, and the same loop-backed program quickly produced an impressive
list of such half-square primes:
17, 47, 71, 283, 881, 1151, 1913, 2591, 3527, 4049,
6047, 7193, 7433, 15137, 20807, 21617, 24197, 26903, 28793, 34847, 46817,
53129, 56443, 69191, 74489, 83231, 84047, 98563, 103049, 103967, 109507,
110441, 112337, 136237, 149057, 151247, 176417, 179999, 198439, 204797, 206081,
235289, 273797, 281249, 290317, 293357, 310463, 321593, 362951, 383683, 388961,
423191, 438047, 470447, 472391, 478241
* * *
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