Monday, 8 September 2014

Near-repdigit prime ZIP codes

Around a decade ago when I was rekindling interest in prime numbers, I studied US ZIP codes because they were noted on the site. I recall from ZIPinfo:
“ZIP code 69577 is not currently assigned by the US Postal Service to any city. Only about 43,000 of the 100,000 possible 5-digit ZIP codes are currently in use.”
and similar sayings for ZIP codes 23333, 86453, 11113, 54342 and no doubt others that I cannot recall. My brother showed me the more user-friendly zipdecode where one could see why the ZIP code 69577 (the number of runs conceded by record-breaking Kent spin bowler “Tich” Freeman in his career) does not exist – if it did it would be in the remote Nebraska Panhandle. In fact no ZIP codes from 69400 to 69999 exist, probably owing to the economic decline of the Plains.

Back in 2004, I looked for near-repdigit prime ZIP codes of the forms RwD and DRw on zipdecode and was struck by how few there actually were relative to the possible ones. I did not count how many possible ZIP codes were but the number I found in use was no more than a handful.

This night, I decided to have a another look and publish my results. Even counting other forms of near-repdigit five-digit prime (RDRw, RRDRR, RwDR) I could find only the following twelve when I found 22229 was no longer used:
ZIP Code City State Notes
10111 New York City New York
11411 Cambria Heights New York a middle-class New York City neighbourhood
16111 Atlantic Pennsylvania in rural western Pennsylvania northeast of Youngstown, Ohio
33331 Fort Lauderdale Florida west of 142nd Avenue and 148th Avenue to 185th Way
38333 Eva Tennessee rural area of Benton County west of Nashville and on Kentucky Lake
44449 North Benton Ohio rural area of Mahoning County south of Berlin Lake and Route 62, due east of Akron.
77377 Tomball Texas Houston exurb in northwest of metropolitan area
77477 Stafford Texas Houston exurb near Sugar Land
77977 Placedo Texas
small village between Victoria and San Antonio Bay
79999 El Paso Texas post office boxes only
94999 Petaluma California southern Napa Valley north of San Pablo Bay. Rural area of Sonoma County and major wine-growing area.
99929 Wrangel Alaska superhumid southeast Alaska not far from Ketchikan
A recent calculation showed thirty-five possible near-repdigit prime ZIP codes without leading zero, which means 34.28 percent of possible near-repdigit prime ZIP codes are in use. Given that the US Postal Service and most others do not need to use every possible code, it is natural that extremely repetitious codes like “two-three-three-three-three” or “two quadruple three” would be avoided, for the obvious reason that they are tough to memorise! Thus, it is not surprising that of the twelve, four are “palindromic wing primes” (out of six possible) and only three have four consecutive copies of one digit. The proportion actually used is certainly larger than I had thought, because it is tough to be really thorough from the prime table I have been using.

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