Alzheimer's disease (AD) is the main cause of dementia and Mild cognitive impairment (MCI) is a prodromal stage of AD whose early detection is considered crucial as it can contribute in slowing the progression of AD. In our study we attempted to classify a subject into AD, MCI, or Healthy Control (HC) groups with the use of electroencephalogram (EEG) data. Due to the time-series nature of EEG we exper-imented with the powerful recurrent neural network (RNN) classifiers and more specifically with models including basic or bidirectional Long Short-Term Memory (LSTM) modules. The EEG signals from 17 channels were preprocessed using a 0.1-32 Hz band-pass filter and then segmented into 2-second epochs during which, the subject had closed eyes. Finally, on each segment Fast Fourier Transform (FFT) was applied. To evaluate our models we studied four different classification problems: problem 1: separating subject into three classes (HC, MCI, AD) and problems 2-4: pairwise classifications AD vs. MCI, AD vs. HC and MCI vs. HC. For each problem we employed two different cross-validation approaches ( a) by segment and (b) by patient. In the first one, segments from a subject EEG may exist in both training and validations set, while in the second one, all the EEG segments of a subject can only exist in either the training or the validation set. In the AD-MCI-HC classification we achieved an accuracy of 99% by segment cross-validation, which was an improvement to earlier studies that utilized recurrent neural network models. In the pairwise classification problems we achieved over 90% accuracy by segment and over 80% by subject.