A309487 Positive integers represented by the quadratic form (the discriminant form) Δ = b^2 - 4ac, where a,b,c are consecutive palindromic primes.

4437, 67088885, 608096563245, 6008043480300405, 60017281285205688005, 600012360124320087600005, 6000055320121974202106400005, 60000010840001925680009488000005, 600000005880000160040000148000000005, 6000000035120000052560000001460000000005

Offset: 1

Author

Philip Mizzi, Sep 06 2019

Keywords

• nonn
• base

Comments

This is an interesting sequence because for most cases Δ<0. The cases where Δ>0 are sparse.

Based on a study of Δ for the case when a,b,c are consecutive primes I conjecture (but have no proof) that now Δ is always negative.

The conjecture in the previous comment is true. It says p(n)^2 <= 4*p(n-1)*p(n+1), and this follows from p(n)^2 <= 4*p(n-1)*p(n), i.e. p(n) <= 4*p(n-1), which is true (see A327447, also Mitrinovic, Sect. VII.18 (b)). - N. J. A. Sloane, Sep 10 2019

The corresponding least palindromic primes are: 11, 929, 98689, 9989899, 999727999, 99999199999, 9999987899999, 999999787999999, ...

Apart from the first term, it appears that the values of "a" and "b" are given by A028990 and A028989, respectively. - Daniel Suteu, Sep 08 2019

Examples

			Consecutive palindromic primes begin with 2,3,5. For a=2, b=3, c=5, Δ=b^2-4ac=-31. Since Δ<0 this is not a member of the sequence.
With consecutive palindromic primes 11,101,131 and a=11, b=101, c=131, Δ=b^2-4ac=4437, the first member of the sequence.
The corresponding values of a,b,c are given in the table bellow.
+----+---------------------+-----------------------+-----------------------+
|  n |          a          |           b           |            c          |
+----+---------------------+-----------------------+-----------------------+
|  1 |                  11 |                   101 |                   131 |
|  2 |                 929 |                 10301 |                 10501 |
|  3 |               98689 |               1003001 |               1008001 |
|  4 |             9989899 |             100030001 |             100050001 |
|  5 |           999727999 |           10000500001 |           10000900001 |
|  6 |         99999199999 |         1000008000001 |         1000017100001 |
|  7 |       9999987899999 |       100000323000001 |       100000353000001 |
|  8 |     999999787999999 |     10000000500000001 |     10000001910000001 |
|  9 |   99999999299999999 |   1000000008000000001 |   1000000032300000001 |
| 10 | 9999999992999999999 | 100000000212000000001 | 100000000252000000001 |
+----+---------------------+-----------------------+-----------------------+
		

References

• D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer.

Crossrefs

Cf. A002385, A327447.

See also A028990 and A028989.

Extensions

a(6)-a(8) from Daniel Suteu, Sep 08 2019

a(9) from Chai Wah Wu, Sep 09 2019

a(10) from Chai Wah Wu, Sep 12 2019