A cubic polynomial with three real roots and rational coefficients can be written in the following form:
(a\(x\) + b) \(\times \) (c\(x\) + d) \( \times \) (e\(x\) + f)
where a, b, c, d, e and f are rational numbers
Let's modify the conditions so that a, b, c, d, e and f are integers (positive or negative) and \(x\) can only take integer values greater than 1. Let's change the \(x\) to an \(n\) so that we have:
(a\(n\) + b) \( \times \) (c\(n\) + d) \( \times \) (e\(n\) + f)
Let's take a specific example where a=1, b=0, c=1, d=1, e=2 and f=3. This gives us:$$n \times (n+1) \times (2n+3)$$As we plug in different values for \(n\), starting with \(n=2\), a series of terms arises. In this case, the terms begin:
42, 108, 220, 390, 630, 952, 1368, 1890, 2530, 3300, 4212, 5278, 6510, 7920, 9520, 11322, 13338, 15580, 18060, 20790, 23782, 27048, 30600, 34450, 38610, 43092, 47908, 53070, 58590, 64480, 70752, 77418, 84490, 91980, 99900, 108262, 117078, 126360, 136120
These terms constitute OEIS A163815 (although the terms for \(n=0\) and \(n=1\) are included). These sorts of sequences involve the multiplication of triple linear combinations of \(n\), in this case \(n\), \(n+1\) and \(2n+3\). This permalink leads to a SageMath algorithm that will generate a sequence of terms for varying values of a, b, c, d, e and f.
If we impose the condition that each of the linear factors must be distinct (and this is the case for the example just shown), then we have a sequence where each member is a sort of pseudo-sphenic number that can be written as a product of three of its divisors but each divisor is a linear combination of an underlying integer. For example, take the number 27048 (my diurnal age yesterday). It can be written as:$$27048 = 23 \times 24 \times 49\\ \text{where } 23=n, 24=n+1 \text{ and } 49=2n+13$$The associated polynomial will cut the \(x\) axis in three locations. Figure 1 shows the situation for \(y=x \times (x+1) \times (2x+3) \) where \(x\)=-1.5, -1 and 0.
Figure 1: Geogebra link |
Let's try another example where a, b, c, d, e, f = 1, -1, 1, 1, 1 , 2. The initial terms generated will be of the form \( (n-1) \times (n+1) \times (n+2) \) and are as follows (starting with \(n=3\) because we want to avoid getting a 1 as a factor):
40, 90, 168, 280, 432, 630, 880, 1188, 1560, 2002, 2520, 3120, 3808, 4590, 5472, 6460, 7560, 8778, 10120, 11592, 13200, 14950, 16848, 18900, 21112, 23490, 26040, 28768, 31680, 34782
This sequence does not appear in the OEIS but it is recognised as being generated from a polynomial (I included the terms for \(n=1\) and \(n=2\) when searching in the OEIS). See Figure 2.
Figure 2 |
Figure 3: permalink |
Figure 4: permalink |
- 106723 = 19 * 41 * 137
- 244807 = 31 * 53 * 149
- 906277 = 61 * 83 * 179
- 1571863 = 79 * 101 * 197
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