Hello. This is Nada from Seito Medical School. The graph below is about prime numbers. This is a numerical example when 3a+5b. If a is a multiple of 5, it becomes 3x5xa+5xb, so it is a multiple of 5 and is not a prime number. Similarly, if b is a multiple of 3, it becomes 3a+5x3xb, which is a multiple of 3 and therefore not a prime number. Also, in a diagonal direction from the prime numbers a and b, the color of the graph advances to the squares diagonally below the prime numbers a and b, but by also advancing to the squares diagonally above, all prime numbers can be covered. From this, 3a+5b is a prime number if at least one of a and b is a prime number, and exceptionally when a of 3a is a multiple of 3 or b of 5b is a multiple of 5. I understand. For example, the diagonally upper row of a=5 is 17, 19, 21, 23. Similarly, the numbers diagonally above a=7 are 23, 25, 27, 29, 31, 33. This leads to the law of prime numbers. Most 3a+5b prime numbers have 3, a, 5, and b relatively prime, and at least one of a and b has a prime value. By the way, mutually prime means that they have no common divisors. My future challenge will be to expand on this and investigate the law of appearance of twin prime numbers.
a\b 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
1 8 13 18 23 28 33 38 43 48 53 58 63 68 73 78 83 88 93 98
2 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101
3 14 19 24 29 34 39 44 49 54 59 64 69 74 79 84 89 94 99 104
4 17 22 27 32 37 42 47 52 57 62 67 72 77 82 87 92 97 102 107
5 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110
6 23 28 33 38 43 48 53 58 63 68 73 78 83 88 93 98 103 108 113
7 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116
8 29 34 39 44 49 54 59 64 69 74 79 84 89 94 99 104 109 114 119
9 32 37 42 47 52 57 62 67 72 77 82 87 92 97 102 107 112 117 122
10 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125
11 38 43 48 53 58 63 68 73 78 83 88 93 98 103 108 113 118 123 128
12 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131
13 44 49 54 59 64 69 74 79 84 89 94 99 104 109 114 119 124 129 134
14 47 52 57 62 67 72 77 82 87 92 97 102 107 112 117 122 127 132 137
15 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140
16 53 58 63 68 73 78 83 88 93 98 103 108 113 118 123 128 133 138 143
17 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146
18 59 64 69 74 79 84 89 94 99 104 109 114 119 124 129 134 139 143 149
19 62 67 72 77 82 87 92 97 102 107 112 117 122 127 132 137 142 147 152
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