Goldbach's Conjecture states every even number can be written as a sum of two primes. This has not yet been proven but it appears highly likely. In fact, it seems that every number greater than 12 can be written as a sum of two primes in at least two different ways. For example:$$\begin{align} 14 &=3 + 11 \\ &= 7+7\\16 &= 3+13 \\ &=5 + 11 \end{align}$$What about expressing primes as linear combinations of pairs of successive primes? Let's say that \( (p, q) \) and \((r, s) \) are pairs each made up of two successive primes. For a given number \(n\) can we find constants \(a, b, c, d\) such that:$$ \begin{align} n &=a \cdot p + b \cdot q \\ &=c \cdot r + d \cdot s \end{align}$$Generally speaking we can and particular examples can be found in OEIS A283392 where \(a=3\), \(b=5\), \(r=5\) and \(s=7\):
A283392 | Integers m of the form \(m = 3 \cdot p + 5 \cdot q = 5 \cdot r + 7 \cdot s \) where \( (p, q) \) and \( (r, s) \) are pairs of consecutive primes. |
Up to 40,000, the members of this sequence are 50, 146, 866, 2162, 4178, 8362, 14372, 17138, 19094, 22504, 25346, 26764, 27544 and 35074 (permalink).
For example:$$ \begin{align} 27544 &=3 \cdot 3433 + 5 \cdot 3449 \\ &= 5 \cdot 2293 + 7 \cdot 2297 \end{align}$$This is equivalent to finding the solutions to the two separate equations with \(x\) and \(y\) necessarily being consecutive primes:$$ \begin{align} 3x_1+5y_1 &= 27544 \text{ with } x_1 = 3433 \text{ and } y_1=3449\\5x_2+7y_2 &=27544 \text{ with }x_2=2293 \text{ and } y_2=2297 \end{align}$$The algorithm that generated the earlier list is easily modified to accommodate different values of \(a, b, c, d\). Let's take the case of \(a=3\), \(b=5\), \(r=7\) and \(s=9\) where the following numbers (not listed in the OEIS) satisfy (permalink):
98, 482, 2314, 8966, 9766, 10562, 12148, 12962, 13066, 14372, 14548, 25588, 31202, 31414, 31846, 34082, 37174, 37588, 39526
An example is 98 where:$$ \begin{align} 98 &=3 \cdot 11 + 5 \cdot 13 \\ &= 7 \cdot 5 + 9\cdot 7 \end{align}$$Of course 98 itself cannot be written as sum of two consecutive primes because:$$ \begin{align} 98 &= 19 + 79 \\ &= 31 + 67 \\ &= 37+61 \end{align} $$Clearly (19, 79), (31, 67) and (37, 61) are not consecutive primes. However, by considering linear combinations of the two consecutive primes, we see (11, 13) and (5, 7) satisfy.
Here is another variation \(a=2\), \(b=3\), \(r=4\) and \(s=5\) where 38 numbers are generated in the range up to 40000. The numbers are (permalink):
407, 541, 1351, 1757, 1783, 2161, 4973, 6077, 6349, 6511, 6889, 7427, 8083, 8723, 10427, 11747, 12751, 13069, 13339, 14833, 18203, 20657, 20791, 21283, 24953, 25111, 29171, 29297, 29323, 31703, 33511, 34157, 35623, 36047, 37397, 37423, 38507, 39181
An example is 407 where:$$ \begin{align} 407 &=2 \cdot 79 + 3 \cdot 83 \\ &= 4 \cdot 43 + 5\cdot 47 \end{align}$$Yet another variations is \(a=3\), \(b=4\), \(r=5\) and \(s=6\) where 23 numbers are generated in the range up to 40000. The numbers are (permalink):
1123, 1157, 4327, 4621, 5671, 5987, 11867, 15877, 19111, 21451, 22157, 23477, 23731, 24257, 26887, 27097, 30199, 30367, 33707, 35773, 36961, 37031, 39607
An example is 1123 where:$$ \begin{align} 1123 &=3 \cdot 157 + 4\cdot 163 \\ &= 5 \cdot 101 + 6\cdot 103 \end{align}$$I'll leave it there but the permalink provides the opportunity to play around with other combinations of \(a, b, c, d\). The number properties described in this post are not base dependent and so are possibly of serious mathematical interests. It seems to be that there should be some practical applications of this technique as a useful way of decomposing or building composite numbers.
Here is an alternative algorithm that generates the same output but uses a different method. It can be used to discover numbers that are linear combinations of two consecutive primes without being equal to a linear combination of two other consecutive primes. For example, it will show that 844 is a linear combination of 3 times 103 and 5 times 107. In other word we have solved the linear equation:$$ 3x+5y = 844$$where \(x\) and \(y\) are prime and \(y\) is the next prime after \(x\).
Obviously, by choosing prime values of \(a, b\) and \(c,d\) we end up with a composite number that is the sum of the products of two pairs of primes. Thus for 844 we have:$$844 = 3 \cdot 103 + 5 \cdot 107$$ which defines the number in the similar way to its prime factorisation:$$844 = 2 \cdot \ 2 \cdot 211$$Unlike the product of a number's prime factors however, this representation is not necessarily unique as we've seen that some numbers have more than one representation as the sum of the products of two pairs of primes e.g. 2162 (permalink):$$ \begin{align} 2162 &=3 \cdot 269 + 5\cdot 271 \\ &= 5 \cdot 179 + 7\cdot 181 \end{align}$$
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