Peptic Ulcer Disease (PUD): Given the patient's history of H. pylori and symptoms of burning abdominal pain, PUD is a highly likely diagnosis. The pain's location in the suprapubic and right lower quadrant (RLQ) areas can be consistent with duodenal or gastric ulcers.

Gastroesophageal Reflux Disease (GERD): Burning abdominal pain, especially if it radiates to the chest, could suggest GERD, which is common in adults and can be exacerbated by H. pylori infection.
Inflammatory Bowel Disease (IBD): Conditions like Crohn's disease or ulcerative colitis could present with abdominal pain, though the absence of fever and diarrhea makes this less likely.
Irritable Bowel Syndrome (IBS): While IBS typically presents with a mix of abdominal pain and alterations in bowel habits, some patients might experience predominant abdominal pain.

Appendicitis: Although the patient does not have fever, appendicitis can present atypically, especially in older adults. The location of pain in the RLQ is a red flag for appendicitis.
Mesenteric Ischemia: This is a vascular emergency that can present with severe abdominal pain out of proportion to physical examination findings. The absence of fever does not rule out this diagnosis.
Diverticulitis: While more common on the left side, diverticulitis can occur on the right and present with pain in the RLQ. The absence of fever and significant abdominal tenderness does not exclude this diagnosis.

Celiac Disease: An autoimmune reaction to gluten that can cause abdominal pain, though it's less common in this age group without a history of malabsorption symptoms.
Intestinal Lymphoma: A rare condition that could present with abdominal pain, though it would typically be accompanied by other systemic symptoms such as weight loss and fever.
Eosinophilic Gastroenteritis: A rare condition characterized by eosinophilic infiltration of the gastrointestinal tract, which can cause abdominal pain, though it's more commonly associated with food allergies or atopic diseases.