Friday, April 12, 2013

Arithmetic sequence of primes

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Question: Suppose a, a+d, a+2d, ..., a+(n-1)d are primes. Find the largest possible n, under the condition that d cannot be a mulitple of 10.

    2     3     5     7    11    13    17    19    23    29
   31    37    41    43    47    53    59    61    67    71
   73    79    83    89    97   101   103   107   109   113
  127   131   137   139   149   151   157   163   167   173
  179   181   191   193   197   199   211   223   227   229
  233   239   241   251   257   263   269   271   277   281
  283   293   307   311   313   317   331   337   347   349
  353   359   367   373   379   383   389   397   401   409
  419   421   431   433   439   443   449   457   461   463
  467   479   487   491   499   503   509   521   523   541
  547   557   563   569   571   577   587   593   599   601
  607   613   617   619   631   641   643   647   653   659
  661   673   677   683   691   701   709   719   727   733
  739   743   751   757   761   769   773   787   797   809
  811   821   823   827   829   839   853   857   859   863
  877   881   883   887   907   911   919   929   937   941
  947   953   967   971   977   983   991   997

(Table of primes up to 1000)




Answer: Lets make a guess. We claim the largest possible n≥5. 

a must be odd, otherwise a+2d is an even number greater than 2 and thus not a prime.

If d is odd, then a+d is an even prime greater than 2 and thus not a prime.

If d is even and not a multiple of 10, then one of a, a+d, a+2d, a+3d, a+4d have the last digit being 5 and hence it must be 5. Note that there is only one odd prime, i.e. 3, which is smaller than 5. Therefore it is only possible if a=5 or a+d=5.

If a+d=5 then a=3, d=2. The sequence is just 3,5,7.

The only case left is that a=5 and then a+5d is a multiple of 5. Therefore n is at most 5.

You can try to search using the table.

5, 11, 17, 23, 29 is a prime sequence with d=6.
5, 17, 29, 41, 53 is a prime sequence with d=12.

Therefore the largest possible n=5.

What if d is allowed to be a multiple of 10?

No matter how large n is, we can construct a sequence a, a+d, a+2d, ..., a+(n-1)d of primes. It is the famous Green-Tao theorem, proved by Ben Green and Terence Tao in 2004.