Update to the count of solutions to the equation
phi ( sigma ( n ) ) = sigma ( phi ( n ) )
In sigma ( phi ( ) ) : From "5" to "5 figures" a table of smallest suitable values of generalized repunit primes based on Fermat prime bases is shown :
smallest suitable values ------ ----------------------------------------------------------------- p3 3, 7, 13, 71, 103, 541, 1091, 1367, 1627 p5 3, 7, 11, 13, 47, 127, 149, 181, 619, 929 p17 3, 5, 7, 11, 47, 71, 419 p257 23, 59 p65537 7, 11
Using the values in that table , the number of known small odd solutions to
phi ( sigma ( n ) ) = sigma ( phi ( n ) )is counted :
cardinalities (so far) exponents solutions #p3 >= 9 9 >= 9 #p5 >= 10 9 * 10 >= 90 #p17 >= 7 9 * 10 * 7 >= 630 #p257 >= 2 9 * 10 * 7 * 2 >= 1260 #p65537 >= 2 9 * 10 * 7 * 2 * 2 >= 2520 ---- total number of small solutions verified (so far) >= 4509
More solutions have now become available :
smallest suitable values updated p3 3 7 13 71 103 541 1091 1367 1627 4177 : 9011 9551 36913 43063 49681 57917 p5 3 7 11 13 47 127 149 181 619 929 3407 : 10949 13241 13873 16519 p17 3 5 7 11 47 71 419 : 4799 p257 23 59 487 967 : 5657 p65537 7 11In this table , the values to the right of the column of colons are not actually proven primes , but are only probable primes .
cardinalities ( so far ) exponents solutions #p3 = 10 10 = 10 #p5 = 11 10 * 11 = 110 #p17 = 7 10 * 11 * 7 = 770 #p257 = 4 10 * 11 * 7 * 4 = 3080 #p65537 = 2 10 * 11 * 7 * 4 * 2 = 6160 ---- total number of small solutions verified ( so far ) = 10130
Counting the probable primes also , this would be the hopeful total count of probable solutions :
cardinalities ( if proven ) exponents solutions #p3 = 16 16 = 16 #p5 = 15 16 * 15 = 240 #p17 = 8 16 * 15 * 8 = 1920 #p257 = 5 16 * 15 * 8 * 5 = 9600 #p65537 = 2 16 * 15 * 8 * 5 * 2 = 19200 ----- total number of small solutions ( if proven ) = 30976
limit of search for probable primes ( so far ) p3 100000 p5 100000 p17 30000 p257 10007 p65537 4423
Additional even solutions can be found in Elaboration of : sigma ( phi ( ) ) : From "5" to "5 figures" .
Appreciation :
Kind thanks are extended to Andy Steward ,
http://www.primes.viner-steward.org/andy/titans.html
http://www.primes.viner-steward.org/andy/annual.html
for the 1st and 2nd lines of this table , to Henri Lifchitz ,
http://www.primenumbers.net/Henri/us/MersFermus.htm
for the 3rd line and to Richard Fischer
http://www.fermatquotient.com/PrimSerien/GenRepu
for the 4th and 5th lines .
Andy Steward is due additional thanks .
2000 Mathematics Subject Classification : Primary 11A25 ; Secondary 11A05 , 11A41 , 11A51 , 11N25 , 11N80 , 11Y11 , 11Y55 , 11Y70
Walter Nissen
originally posted 2008-06-08 link added 2008-10-15