In the Year 2525

   The song wonders, “In the year 2525, if man is still alive . . . ”

   In what year must man be alive to hear the cicadas and see Halley’s comet in the same year?

Background:
   The cicadas last appeared in 2004 and return every 17 years.  
   Halley's Comet last appeared in 1986 and returns every 76 years.

Answer:
3126 is the next year they will both appear together.

Note:
   The last time they appeared together was in 1834.

 

 

Brute Force:
   Create two series in Excel for the two events. Then creatively prune the columns and visually find entries in the two columns that match.

Diophantine Solution:
   Given that 1986 + 76i = 1987 + 17j
   Simplifying 76i –17j = 1.
   See Diophantine example for technique used to solve equation.
   i = 15 + 17m where m = any integer
   j = 67 + 76m where m = any integer
   When m = 0, i = 15 and j = 67. Halley's comet will appear in 1986 + 76 * 15 = 3126

Doublecheck:
   1987 + 17 * 67 = 3126. The cicadas will appear that year too.