[ May 21, 2007 ]
A random walk around the powers of ten
A conjecture of Milton L. Brown (email)

Let 10ⁿ+x be the smallest PRP (prime) past 10ⁿ
let 10ⁿ–y be the largest PRP (prime) before 10ⁿ

Then f(n) = x–y and F(n) = sum f(m) for m = 1 to n

F(n) is negative for values of n in the first set of brackets,
and positive for values of n in the second set of brackets below.

F(n) remains negative for after n = 2652

[2653,*] where * at 4999 the sum is -653126

The conjecture is that it is always negative thereafter.

[1,2]------ [3] [4,22]----- [23] [24,28]---- [29,33] [34]------- [35] [36,39]---- [40] [41,45]---- [46,53] [54,239]--- [240] [241,303]-- [304,425] [426]------ [427] [428]------ [429,532] [533]------ [534,554] [555]------ [556,573] [574,575]---[576,1060] [1061,1065] [1066] [1067,1074] [1075,1082] [1083,1091] [1092,1114] [1115,1127] [1128,1144] [1145]----- [1146,1391] [1392,1400] [1401,1402] [1403,1411] [1412,1414] [1415,1455] [1456,1459] [1460]----- [1461,1472] [1473,1484] [1485,1681] [1682,1761] [1762,1766] [1767]----- [1768,1786] [1787]----- [1788,1817] [1818,1830] [1831,2222] [2223,2283] [2284,2286] [2287,2292] [2293,2305] [2306,2310] [2311,2312] [2313,2382] [2383,2384] [2385]----- [2386,2393] [2394,2403] [2404,2407] [2408,2413] [2414] [2415,2416] [2417,2423] [2424,2464] [2465,2468] [2469,2473] [2474,2606] [2607]----- [2608,2610] [2611]----- [2612,2613] [2614,2631] [2632,2633] [2634,2638] [2639,2640] [2641]----- [2642,2652] [2653,*] * at 4999 the sum is -653126 * at 6300 the sum is -940962

Also, Milton provides some graphs for handy visualisation,
They show the decrease into 1000 unit steps.

diff1000.docx
diff2000.docx
diff3000.docx
diff4000.docx
diff5000.docx
diff6000.docx