Rose Peltesohn

Solution of Heffter's Difference Problems

Peltesohn solved the Difference Problems of Lothar Heffter (1896) in combinatorics in 1939. A Difference Triple (a, b, c) is defined as three different elements from the set ${\displaystyle 1,2,\dots ,v-1}$, whose sum ${\displaystyle modv}$ equals zero (${\displaystyle a+b+c=0modv}$) or for which one element ${\displaystyle modv}$ equals the sum of the other two (${\displaystyle a+b=cmodv}$).

• First Difference Problem of Heffter: Let ${\displaystyle v=6m+1}$. Is there a partition of the set ${\displaystyle 1,2,\dots ,3m}$ in difference triples?
• Second Difference Problem of Heffter: Let ${\displaystyle v=6m+3}$. Is there a partition of the set ${\displaystyle 1,2,\dots ,2m,2m+2,\dots ,3m+1}$ in difference triples ?

Following Peltesohn, such a partition exists with the exception of the case v = 9.

An example of the partition for ${\displaystyle v=13}$ is: ${\displaystyle (1,3,4)}$ (with ${\displaystyle 1+3=4mod13}$) and ${\displaystyle (2,5,6)}$ (with ${\displaystyle 2+5+6=13=0mod13}$).

The solution of the Difference Problem of Heffter also gives a construction of cyclic Steiner triple systems.

Literature

• Maximilian Pinl (1969). "Kollegen in einer dunklen Zeit". Jahresbericht Deutsche Mathematikervereinigung (DMV). 71: 171–228. Archived from the original on 2014-11-07. Retrieved 2014-11-06. in particular pages 188–189