Cullen and Woodall numbers

Cullen numbers are given by the expression n.2ⁿ + 1 and Woodall numbers by n.2ⁿ - 1. They are often abbreviated to C(n) and W(n) respectively. A generalization of Cullen and Woodall numbers is straightforward.

The tables of Cullen and Woodall numbers abide by my standard format and include factorizations for index n up to 1000.

Anyone who wishes to contribute to the factorization of Cullen and Woodall numbers is welcome to download the tables given above. Resulting factorizations should be reported to me and I'll record contributions in the progress file.

To give an idea of the expected difficulty of finding new factors, both of the tables have been swept by ECM to B1=3,000,000 or about 40 digits. Some entries have had substantially more ECM effort applied, especially the Most Wanted Numbers. As can be seen from the tables below, all cofactors have at least 141 digits and both tables have been completed to an index of 660.

Factoring the Cullen and Woodall numbers by NFS has now become sufficiently difficult that it would be wise to check with me that your intended target is neither factored nor in progress before embarking on a lengthy computation.

The current most wanted list is

Woodall(951) is the only number in the tables with no known factors;  The other most wanted numbers are all the composite cofactors under 512 bits.

The first five holes in the tables are:

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