10) Non-primes
to the function x2 + xy + y2 (when x and y are positive
integers) appear to follow a pattern, and numbers that are not solutions to x2
+ xy + y2 (when x and y are integers) also appear to follow a
pattern.
In
multiplication, a prime number can only be expressed as a product of two
positive integers in one way. For
example, 7 is a prime number because it can only be expressed as a product of
two positive integers in one way: 1∙7 = 7.
If we apply this definition of primes to the function of x2 +
xy + y2, then prime numbers can only be expressed as x2 +
xy + y2 in one way where x and y are both positive integers. Using this definition of prime numbers to x2
+ xy + y2, most solutions to x2 + xy + y2 are
prime numbers. For example, 19 is a
prime number in x2 + xy + y2 because (2, 3) = 19. (According to the equalities in Point 7, 19
can also be expressed as (3, 2), (-2, -3), (-3, -2), (2, -5), (-5, 2), (-2, 5),
(5, -2), (-5, 3), (3, -5), (5, -3) and (-3, 5), but none of these count because
of either the commutative property or negative numbers.) However, some numbers are not primes to the
function x2 + xy + y2, because they can be expressed as x2
+ xy + y2 by more than two different positive integers. For example, 91 can be expressed as (1, 9)
and (5, 6), and 133 can be expressed as (1, 11) and (4, 9).
1
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2
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3
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4
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5
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6
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7
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8
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9
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10
|
|
1
|
3
|
7
|
13
|
21
|
31
|
43
|
57
|
73
|
91
|
111
|
2
|
7
|
12
|
19
|
28
|
39
|
52
|
67
|
84
|
103
|
124
|
3
|
13
|
19
|
27
|
37
|
49
|
63
|
79
|
97
|
117
|
139
|
4
|
21
|
28
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37
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48
|
61
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76
|
93
|
112
|
133
|
156
|
5
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31
|
39
|
49
|
61
|
75
|
91
|
109
|
129
|
151
|
175
|
6
|
43
|
52
|
63
|
76
|
91
|
108
|
127
|
148
|
171
|
196
|
7
|
57
|
67
|
79
|
93
|
109
|
127
|
147
|
169
|
193
|
219
|
8
|
73
|
84
|
97
|
112
|
129
|
148
|
169
|
192
|
217
|
244
|
9
|
91
|
103
|
117
|
133
|
151
|
171
|
193
|
217
|
243
|
271
|
10
|
111
|
124
|
139
|
156
|
175
|
196
|
219
|
244
|
271
|
300
|
In fact, if x2
+ xy + y2 is a product of two other numbers in the form of x2
+ xy + y2 where x ≠ y, then it appears that that number is not
prime. So 91 is not prime because 91 =
7∙13, and 7 = (1, 2) where 1 ≠ 2 and 13 = (1, 3) where 1 ≠ 3. Similarly, 133 is not prime because 133 =
7∙19, and 7 = (1, 2) where 1 ≠ 2 and 19 = (2, 3) where 2 ≠ 3. (Also note that 21 is prime even though 21 =
3∙7, because 3 = (1, 1) where 1 = 1.)
This can be explained using two of the formulas obtained from Point 9,
that (a, b) ∙ (c, d) = (-ac + bd, ac + ad + bc) and (a, b) ∙ (c, d) = (ac + bc
+ bd, ad – bc), because if a ≠ b then -ac + bd and ad – bc are two unique
answers. For example, in our 91 example,
(1, 2) ∙ (1, 3) = (-1∙1 + 2∙3, 1∙1 + 1∙3 + 2∙1) = (5, 6) and (1, 2) ∙ (1, 3) =
(1∙1 + 2∙1 + 2∙3, 1∙3 – 2∙1) = (1, 9).
However, in our 21 example, (1, 1) ∙ (1, 2) would make a = b, which
would make –ac + bd = ad – bc, therefore giving two equal answers (1, 4) and
(1, 4), which makes 21 not prime.
There are
also numbers that are not solutions to x2 + xy + y2 (when
x and y are integers), which do not appear on the number table at all. We shall call these numbers zero-primes of x2
+ xy + y2. Zero-primes of x2
+ xy + y2 include 2, 5, 6, 8, 10, 11, 14, 15, 17, 18, 20, etc. Any number in the form of 3n + 2, where n is
any positive integer, is a zero-prime, such as 2, 5, 8, 11, 14, 17, etc. We can show this inductively from the number
table, if we change all solutions in terms of mod 3, and see that every 3 x 3
square repeats itself infinitely and there is no solution of 2, meaning there
are no solutions in the form of 3n + 2.
1
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2
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3
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4
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5
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6
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7
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8
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9
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10
|
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1
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0
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1
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1
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0
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1
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1
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0
|
1
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1
|
0
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2
|
1
|
0
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1
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1
|
0
|
1
|
1
|
0
|
1
|
1
|
3
|
1
|
1
|
0
|
1
|
1
|
0
|
1
|
1
|
0
|
1
|
4
|
0
|
1
|
1
|
0
|
1
|
1
|
0
|
1
|
1
|
0
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5
|
1
|
0
|
1
|
1
|
0
|
1
|
1
|
0
|
1
|
1
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6
|
1
|
1
|
0
|
1
|
1
|
0
|
1
|
1
|
0
|
1
|
7
|
0
|
1
|
1
|
0
|
1
|
1
|
0
|
1
|
1
|
0
|
8
|
1
|
0
|
1
|
1
|
0
|
1
|
1
|
0
|
1
|
1
|
9
|
1
|
1
|
0
|
1
|
1
|
0
|
1
|
1
|
0
|
1
|
10
|
0
|
1
|
1
|
0
|
1
|
1
|
0
|
1
|
1
|
0
|
But 6 is
also a zero-prime, but not in the form of 3n + 2. So zero-primes can also include numbers in
the form of 4n + 2, which again we can also show inductively on the number
table in terms of mod 4 because every 4 x 4 square repeats infinitely and there
is no solution of 2.
1
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2
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3
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4
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5
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6
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7
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8
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9
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10
|
|
1
|
3
|
3
|
1
|
1
|
3
|
3
|
1
|
1
|
3
|
3
|
2
|
3
|
0
|
3
|
0
|
3
|
0
|
3
|
0
|
3
|
0
|
3
|
1
|
3
|
3
|
1
|
1
|
3
|
3
|
1
|
1
|
3
|
4
|
1
|
0
|
1
|
0
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1
|
0
|
1
|
0
|
1
|
0
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5
|
3
|
3
|
1
|
1
|
3
|
3
|
1
|
1
|
3
|
3
|
6
|
3
|
0
|
3
|
0
|
3
|
0
|
3
|
0
|
3
|
0
|
7
|
1
|
3
|
3
|
1
|
1
|
3
|
3
|
1
|
1
|
3
|
8
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
1
|
0
|
9
|
3
|
3
|
1
|
1
|
3
|
3
|
1
|
1
|
3
|
3
|
10
|
3
|
0
|
3
|
0
|
3
|
0
|
3
|
0
|
3
|
0
|
In fact, it
appears that most numbers in the form of (3m + 2)2n + k(3m + 2) when
m, n, and k are integers and 0 < k < 3m + 2 are zero-prime, including 25n
+ 5 (m = 1, k = 1), 25n + 10 (m = 1, k = 2), 25n + 15 (m = 1, k = 3), 25n + 20
(m = 1, k = 4), 64n + 8 (m = 2, k = 1), 64n + 24 (m = 2, k = 3), 64n + 32 (m =
2, k = 4), 64n + 40 (m = 2, k = 5), 64n + 56 (m = 2, k = 7), etc. However, some exceptions occur to this rule
at 64n + 16 (m = 2, k = 2) because (0, 4) = 16 and also at 64n + 48 (m = 2, k =
6) because (4, 4) = 48. Other exceptions
may also occur with larger numbers, but the exact pattern is unknown to me.
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