The integers are 37 and 39.
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There are no two consecutive integers that sum to 58. With two consecutive integers, one is even, the other is odd. The sum of an even number and an odd number is odd. 58 is even so cannot be the sum of two consecutive integers.
The numbers are 55, and 57. Two consecutive integers have an odd sum.
The two consecutive negative odd integers having 74 as the sum of their squares are -5 and -7.
29 cannot be the sum of two odd integers. The sum of two odd integers is always an even integer.
There is a set of two consecutive integers that have a sum of 123; one odd and one even. They are 61 and 62.