On Cullen numbers which are both Riesel and Sierpiński numbers

Abstract

We describe an algorithm to determine whether or not a given system of congruences is satisfied by Cullen numbers. We use this algorithm to prove that there are infinitely many Cullen numbers which are both Riesel and Sierpiński. (Such numbers should be discarded if you are searching prime numbers with Proth's theorem.)

Generalized Woodall Numbers: An Investigation of Properties of Woodall and Cullen Numbers via Their Third Order Linear Recurrence Relations

“…The first few Woodall numbers are: 1,7,23,63,159,383,895,2047,4607,10239,22527,49151,106495,229375, 491519, 1048575, . .…”

“…The first few Cullen numbers are: 1,3,9,25,65,161,385,897,2049,4609,10241,22529,49153,106497,229377,491521, ... (sequence A002064 in the OEIS). Woodall and Cullen sequences have been studied by many authors and more detail can be found in the extensive literature dedicated to these sequences, see for example, [1,2,6,9,10,11,13,15,16,17,18] and references therein. Note that {R n } and {C n } hold the following relations:…”

Cullen numbers in sums of terms of recurrence sequence

“…For instance, Luca and Shparlinski [9] studied on the pseudoprime Cullen numbers and Berrizbeitia et. al., [2] investigated on Cullen numbers which are both Riesel and Sierpinski numbers. Further, Luca and Stȃnicȃ [10] proved that there are only finitely many Cullen numbers in a binary recurrence sequence under some additional assumptions.…”

Generalized Cullen numbers in linear recurrence sequences

“…A Cullen number is a number of the form m2 m + 1 (denoted by C m ), where m is a nonnegative integer. A few terms of this sequence are 1,3,9,25,65,161,385,897,2049,4609, 10241, 22529, . .…”

“…The problem of finding Cullen numbers belonging to others known sequences has attracted much attention in the last two decades. We cite [21] for pseudoprime Cullen numbers, and [1] for Cullen numbers which are both Riesel and Sierpiński numbers.…”