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Encyclopedia > Leyland number

A Leyland number is an integer of the form xy + yx with 1 < xy. The first few Leyland numbers are

8, 17, 32, 54, 57, 100, 145, 177, 320, 368, 512, 593, 945 (sequence A076980 in OEIS) 8 (eight) is the natural number following 7 and preceding 9. ... 17 (seventeen) is the natural number following 16 and preceding 18. ... 32 (thirty-two) is the natural number following 31 and preceding 33. ... Fifty-four (54) is the natural number following 53 and preceding 55. ... 57 is the natural number following 56 and preceding 58. ... 100 (one hundred) (the Roman numeral is C for centum) is the natural number following 99 and preceding 101. ... 145 is the natural number following 144 and preceding 146. ... Cardinal one hundred and seventy-seven Ordinal 176th (one hundred and seventy-seventh) Factorization Divisors 1, 3, 59, 177 Roman numeral CLXXVII Binary 10110001 Hexadecimal B1 177 is an odd natural number between 176 and 178. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

Because of the commutative property of addition, it is not actually required that y be greater than or equal to x, this simply keeps the algorithm from getting bogged down with duplicates. The requirement that x and y both be greater than 1, however, is important, since without it every positive integer would be a Leyland number of the form 1y + y1.

Leyland numbers that are also prime are listed in A094133. The largest known Leyland number that is also a prime is 26384405 + 44052638.


  • Richard Crandall and Carl Pomerance, Prime Numbers : A Computational Perspective, Springer, 2005



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