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Using the ZEUS detector at HERA, we have studied the reaction e~+ p -> e~+ X for Q~2 > 5000 GeV~2 with a 20.1 pb~{-1} data sample collected during the years 1994 to 1996. For Q~2 below 15000 GeV~2, the data are in good agreement with Standard Model expectations. For Q~2 > 35000 Gev~2, two events are observed while 0.145+-0.013 events are expected. A statistical analysis of a large ensemble of simulated Standard Model experiments indicates that with probability 6.0%, an excess at least as unlikely as that observed would occur above some Q~2 cut. For x>0.55 and y>0.25, four events are observed where 0.91+-0.08 events are expected. A statistical analysis of the two-dimensional distribution of the events in x and y yields a probability of 0.72% for the region x>0.55 and y>0.25 and a probability of 7.8% for the entire Q~2 > 5000 GeV~2 data sample. The observed excess above Standard Model expectations is particularly interesting because it occurs in a previously unexplored kinematic region

Comparison of ZEUS data with standard model predictions for $e^+p \rightarrow e^+X$ scattering at high $x$ and $Q^2$