Znám's Problem

Author

Eric W. Weisstein
September 18, 2002

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumberTheory/ZnamsProblem.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ZnamsProblem.html.

Additional Authors

Data from Margherita Barile
September 2002

Data

Znam[5]:={{2,3,7,47,395},{2,3,11,23,31}}
Znam[6]:={{2,3,7,43,1823,193667},{2,3,7,47,403,19403},{2,3,7,47,415,8111},{2,3,7,47,583,1223},{2,3,7,55,179,24323}}
Znam[7]:={{2,3,7,43,1807,3263447,2130014000915},{2,3,7,43,1807,3263591,71480133827},{2,3,7,43,1807,3264187,14298637519},{2,3,7,43,3559,3667,33816127},{2,3,7,47,395,779831,6020372531},{2,3,7,67,187,283,334651},{2,3,11,17,101,149,3109},{2,3,11,23,31,47063,442938131},{2,3,11,23,31,47095,59897203},{2,3,11,23,31,47131,30382063},{2,3,11,23,31,47243,12017087},{2,3,11,23,31,47423,6114059},{2,3,11,23,31,49759,866923},{2,3,11,23,31,60563,211031},{2,3,11,31,35,67,369067}}
Znam[9]:={{2,3,7,43,1807,3263443,10650056950807,113423713055421844361000447,2572987736655734348107429290411162753668127385839515}};
Znam[10]:={{2,3,11,23,31,47059,2214502423,4904020979258368507,24049421765006207593444550012151040547,115674937446230858658157460659985774139375256845351399814552547262816571295}}
Znam[13]:={{3,4,5,7,29,41,67,89701,230865947737,5726348063558735709083,172509500849902989281836693100308633431804359,428596832385786878003379724333422434733374132288492887588514414196525736338894723726187,4754717350939481607957800419492385085075824146211772113509986641996660938291241829057128592157690457223767635578617952597534252744929680110197649133558287624332177380360519}}

Code

ZnamQuotients[l_List?VectorQ]:=Module[{n=Length[l],i},
Table[(1+Times@@Drop[l,{i}])/l[[i]],{i,n}]
]

ZnamQ[l_List?VectorQ]:=IntegerQ/@And@@ZnamQuotients[l]

©2005 Wolfram Research, Inc. except for portions noted otherwise

Checks

znams=Znam/@(#[[1,1,1]]&/@DownValues[Znam]);

Map[Length,znams,{2}]

Map[ZnamQ,znams,{2}]

Length/@IntegerDigits/@Znam[13][[1]]

Egyptian Fractions

Map[Plus@@(1/#)+1/Times@@#&,znams,{2}]//Simplify

v5.0

Reduce[Plus@@(1/Array[x,5])+1/Times@@Array[x,5]==1&&Alternatives@@Array[x,5]∈Integers,Array[x,5]]//Timing

Reduce[Plus@@(1/Array[x,5])+1/Times@@Array[x,5]==1&&Alternatives@@Array[x,5]∈Integers&&
And@@(0<#<10^5&/@Array[x,5]),
Array[x,5]]//Timing

Sloane

<<EISFormat`

FormatSequence[Join[{0,0,0},Length/@Table[Znam[n],{n,5,7}],{93}],075441,2,
Name->"a(n) is the number of solutions to the n-term Znam's problem.",
html["Znam's Problem"],
Cross->{75461}
]

FormatSequence[Flatten[Table[Sort[Sort/@Znam[n]],{n,5,7}]],75461,1,
Name->"Sorted list of solutions to the n-term Znam's problem.",
html["Znam's Problem"],
Comment->"Number of n-term solutions is given by A075441.",
Example->"The 5-term solutions are 2,3,7,47,395; 2,3,11,23,31.  Followed by the 5 6-term solutions, etc.",
Cross->{075441}
]